Hall Effect Navigation

Introduction
The Hall Effect

· evolution of resistance concepts

· the Hall effect and the Lorentz force

· the van der Pauw technique

Resistivity and Hall Measurements

· sample geometry

· definitions for resistivity measurements

· resistivity measurements

· resistivity calculations

· definitions for Hall measurements

· Hall measurements

· Hall calculations

· Sample Hall Worksheet

· Worksheet with Typical Data

Algorithm
References

I. Introduction

The objective of this Web site is twofold: (1) to describe the Hall measurement technique for determining the carrier density and mobility in semiconductor materials and (2) to initiate an electronic interaction forum where workers interested in the Hall effect can exchange ideas and information. The following pages will lead the reader through an introductory description of the Hall measurement technique, covering basic principles, equipment, and recommended procedures.

The importance of the Hall effect is underscored by the need to determine accurately carrier density, electrical resistivity, and the mobility of carriers in semiconductors. The Hall effect provides a relatively simple method for doing this. Because of its simplicity, low cost, and fast turnaround time, it is an indispensable characterization technique in the semiconductor industry and in research laboratories. In a recent industrial survey, it is listed as one of the most-commonly used characterization tools. Furthermore, two recent Nobel prizes (1985, 1998) are based upon the Hall effect.

The history of the Hall effect begins in 1879 when Edwin H. Hall discovered that a small transverse voltage appeared across a current-carrying thin metal strip in an applied magnetic field. Until that time, electrical measurements provided only the carrier density-mobility product, and the separation of these two important physical quantities had to rely on other difficult measurements. The discovery of the Hall effect removed this difficulty. Development of the technique has since led to a mature and practical tool, which today is used routinely for testing the electrical properties and quality of almost all of the semiconductor materials used by industry.

II The Hall Effect

Evolution of Resistance Concepts

The Hall Effect and the Lorentz Force

The Vander Pauw method

Evolution of Resistance Concepts

Electrical characterization of materials evolved in three levels of understanding. In the early 1800s, the resistance R and conductance G were treated as measurable physical quantities obtainable from two-terminal I-V measurements (i.e., current I, voltage V). Later, it became obvious that the resistance alone was not comprehensive enough since different sample shapes gave different resistance values. This led to the understanding (second level) that an intrinsic material property like resistivity (or conductivity) is required that is not influenced by the particular geometry of the sample. For the first time, this allowed scientists to quantify the current-carrying capability of the material and carry out meaningful comparisons between different samples.

By the early 1900s, it was realized that resistivity was not a fundamental material parameter, since different materials can have the same resistivity. Also, a given material might exhibit different values of resistivity, depending upon how it was synthesized. This is especially true for semiconductors, where resistivity alone could not explain all observations. Theories of electrical conduction were constructed with varying degrees of success, but until the advent of quantum mechanics, no generally acceptable solution to the problem of electrical transport was developed. This led to the definitions of carrier density n and mobility µ (third level of understanding) which are capable of dealing with even the most complex electrical measurements today.

The Hall Effect and the Lorentz Force

The basic physical principle underlying the Hall effect is the Lorentz force. When an electron moves along a direction perpendicular to an applied magnetic field, it experiences a force acting normal to both directions and moves in response to this force and the force effected by the internal electric field. For an n-type, bar-shaped semiconductor shown in Fig.1, the carriers are predominately electrons of bulk density n. We assume that a constant current I flows along the x-axis from left to right in the presence of a z-directed magnetic field. Electrons subject to the Lorentz force initially drift away from the current line toward the negative y-axis, resulting in an excess surface electrical charge on the side of the sample. This charge results in the Hall voltage, a potential drop across the two sides of the sample. (Note that the force on holes is toward the same side because of their opposite velocity and positive charge.) This transverse voltage is the Hall voltage V_H and its magnitude is equal to IB/qnd, where I is the current, B is the magnetic field, d is the sample thickness, and q (1.602 x 10^-19 C) is the elementary charge. In some cases, it is convenient to use layer or sheet density (n_s = nd) instead of bulk density. One then obtains the equation

n_s = IB/q|V_H|.

(1)

Thus, by measuring the Hall voltage V_H and from the known values of I, B, and q, one can determine the sheet density n_s of charge carriers in semiconductors. If the measurement apparatus is set up as described later in Section III, the Hall voltage is negative for n-type semiconductors and positive for p-type semiconductors. The sheet resistance R_S of the semiconductor can be conveniently determined by use of the van der Pauw resistivity measurement technique. Since sheet resistance involves both sheet density and mobility, one can determine the Hall mobility from the equation

µ = |V_H|/R_SIB = 1/(qn_SR_S).

(2)

If the conducting layer thickness d is known, one can determine the bulk resistivity (r = R_Sd) and the bulk density (n = n_S/d).

The van der Pauw Technique

In order to determine both the mobility µ and the sheet density n_s, a combination of a resistivity measurement and a Hall measurement is needed. We discuss here the van der Pauw technique which, due to its convenience, is widely used in the semiconductor industry to determine the resistivity of uniform samples (References 3 and 4). As originally devised by van der Pauw, one uses an arbitrarily shaped (but simply connected, i.e., no holes or nonconducting islands or inclusions), thin-plate sample containing four very small ohmic contacts placed on the periphery (preferably in the corners) of the plate. A schematic of a rectangular van der Pauw configuration is shown in Fig. 2.

The objective of the resistivity measurement is to determine the sheet resistance R_S. Van der Pauw demonstrated that there are actually two characteristic resistances R_A and R_B, associated with the corresponding terminals shown in Fig. 2. R_A and R_B are related to the sheet resistance R_S through the van der Pauw equation

exp(-pR_A/R_S) + exp(-pR_B/R_S) = 1

(3)

which can be solved numerically for R_S.

The bulk electrical resistivity r can be calculated using

r = R_Sd.

(4)

To obtain the two characteristic resistances, one applies a dc current I into contact 1 and out of contact 2 and measures the voltage V₄₃ from contact 4 to contact 3 as shown in Fig. 2. Next, one applies the current I into contact 2 and out of contact 3 while measuring the voltage V₁₄ from contact 1 to contact 4. R_A and R_B are calculated by means of the following expressions:

R_A = V₄₃/I₁₂ and R_B = V₁₄/I₂₃.

(5)

The objective of the Hall measurement in the van der Pauw technique is to determine the sheet carrier density n_s by measuring the Hall voltage V_H. The Hall voltage measurement consists of a series of voltage measurements with a constant current I and a constant magnetic field B applied perpendicular to the plane of the sample. Conveniently, the same sample, shown again in Fig. 3, can also be used for the Hall measurement. To measure the Hall voltage V_H, a current I is forced through the opposing pair of contacts 1 and 3 and the Hall voltage V_H (= V₂₄) is measured across the remaining pair of contacts 2 and 4. Once the Hall voltage V_H is acquired, the sheet carrier density n_s can be calculated via n_s = IB/q|V_H| from the known values of I, B, and q.

There are practical aspects which must be considered when carrying out Hall and resistivity measurements. Primary concerns are (1) ohmic contact quality and size, (2) sample uniformity and accurate thickness determination, (3) thermomagnetic effects due to nonuniform temperature, and (4) photoconductive and photovoltaic effects which can be minimized by measuring in a dark environment. Also, the sample lateral dimensions must be large compared to the size of the contacts and the sample thickness. Finally, one must accurately measure sample temperature, magnetic field intensity, electrical current, and voltage.

III. Resistivity and Hall Measurements

Sample Geometry

Definitions for Resistivity Measurements
Resistivity Measurements
Resistivity Calculations
Definitions for Hall Measurements
Hall Measurements

Hall Calculations

The following procedures for carrying out Hall measurements provide a guideline for the beginning user who wants to learn operational procedures, as well as a reference for experienced operators who wish to invent and engineer improvements in the equipment and methodology.

Sample Geometry

It is preferable to fabricate samples from thin plates of the semiconductor material and to adopt a suitable geometry, as illustrated in Fig. 4. The average diameters (D) of the contacts, and sample thickness (d) must be much smaller than the distance between the contacts (L). Relative errors caused by non-zero values of D are of the order of D/L.

The following equipment is required:

Permanent magnet, or an electromagnet (500 to 5000 gauss)
Constant-current source with currents ranging from 10 µA to 100 mA (for semi-insulating GaAs, ρ ~ 10⁷ Ω·cm, a range as low as 1 nA is needed)
High input impedance voltmeter covering 1 µV to 1 V
Sample temperature-measuring probe (resolution of 0.1 °C for high accuracy work)

Definitions for Resistivity Measurements

Four leads are connected to the four ohmic contacts on the sample. These are labeled 1, 2, 3, and 4 counterclockwise as shown in Fig. 4a. It is important to use the same batch of wire for all four leads in order to minimize thermoelectric effects. Similarly, all four ohmic contacts should consist of the same material.

We define the following parameters (see Fig. 2):

ρ = sample resistivity (inΩ·cm)

d = conducting layer thickness (in cm)

I₁₂ = positive dc current I injected into contact 1 and taken out of contact 2. Likewise for I₂₃, I₃₄, I₄₁, I₂₁, I₁₄, I₄₃, I₃₂(in amperes, A)

V₁₂ = dc voltage measured between contacts 1 and 2 (V₁ - V₂) without applied magnetic field (B = 0). Likewise for V₂₃, V₃₄, V₄₁, V₂₁, V₁₄, V₄₃, V₃₂ (in volts, V)

Resistivity Measurements

The data must be checked for internal consistency, for ohmic contact quality, and for sample uniformity.

Set up a dc current I such that when applied to the sample the power dissipation does not exceed 5 mW (preferably 1 mW). This limit can be specified before the automatic measurement sequence is started by measuring the resistance R between any two opposing leads (1 to 3 or 2 to 4) and setting

I < (200R)^-0.5.

(6)

Apply the current I₂₁ and measure voltage V₃₄
Reverse the polarity of the current (I₁₂) and measure V₄₃
Repeat for the remaining six values (V₄₁, V₁₄, V₁₂, V₂₁, V₂₃, V₃₂)

Eight measurements of voltage yield the following eight values of resistance, all of which must be positive:

	R_21,34 = V₃₄/I₂₁, R_12,43 = V₄₃/I₁₂,
	R_32,41 = V₄₁/I₃₂, R_23,14 = V₁₄/I₂₃,		(7)
	R_43,12 = V₁₂/I₄₃, R_34,21 = V₂₁/I₃₄,
	R_14,23 = V₂₃/I₁₄, R_41,32 = V₃₂/I₄₁.

Note that with this switching arrangement the voltmeter is reading only positive voltages, so the meter must be carefully zeroed.

Because the second half of this sequence of measurements is redundant, it permits important consistency checks on measurement repeatability, ohmic contact quality, and sample uniformity.

Measurement consistency following current reversal requires that:

	R_21,34 = R_12,43	R_43,12 = R_34,21
	R_32,41 = R_23,14	R_14,23 = R_41,32		(8)

The reciprocity theorem requires that:

	R_21,34 + R_12,43 = R_43,12 + R_34,21, and
	R_32,41 + R_23,14 = R_14,23 + R_41,32.		(9)

If any of the above fail to be true within 5 % (preferably 3 %), investigate the sources of error.

Resistivity Calculations

The sheet resistance R_S can be determined from the two characteristic resistances

	R_A = (R_21,34 + R_12,43 + R_43,12 + R_34,21)/4 and
	R_B = (R_32,41 + R_23,14 + R_14,23 + R_41,32)/4		(10)

via the van der Pauw equation [Eq. (3)]. For numerical solution of Eq. (3), see the routine in Section IV. If the conducting layer thickness d is known, the bulk resistivity ρ = R_S d can be calculated from R_S.

Definitions for Hall Measurements

The Hall measurement, carried out in the presence of a magnetic field, yields the sheet carrier density n_s and the bulk carrier density n or p (for n-type or p-type material) if the conducting layer thickness of the sample is known. The Hall voltage for thick, heavily doped samples can be quite small (of the order of microvolts).

The difficulty in obtaining accurate results is not merely the small magnitude of the Hall voltage since good quality digital voltmeters on the market today are quite adequate. The more severe problem comes from the large offset voltage caused by nonsymmetric contact placement, sample shape, and sometimes nonuniform temperature.

The most common way to control this problem is to acquire two sets of Hall measurements, one for positive and one for negative magnetic field direction. The relevant definitions are as follows (Fig. 3):

I₁₃ = dc current injected into lead 1 and taken out of lead 3. Likewise for I₃₁, I₄₂, I₂₄.

B = constant and uniform magnetic field intensity (to within 3 %) applied parallel to the z-axis within a few degrees (Fig .3). B is positive when pointing in the positive z direction, and negative when pointing in the negative z direction.

V_24P = Hall voltage measured between leads 2 and 4 with magnetic field positive for I₁₃. Likewise for V_42P, V_13P, and V_31P.

Similar definitions for V_24N, V_42N, V_13N, V_31N apply when the magnetic field B is reversed.

Hall Measurements

The procedure for the Hall measurement is:

Apply a positive magnetic field B
Apply a current I₁₃ to leads 1 and 3 and measure V_24P
Apply a current I₃₁ to leads 3 and 1 and measure V_42P
Likewise, measure V_13P and V_31P with I₄₂ and I₂₄, respectively
Reverse the magnetic field (negative B)
Likewise, measure V_24N, V_42N, V_13N, and V_31N with I₁₃, I₃₁, I₄₂, and I₂₄, respectively

The above eight measurements of Hall voltages V_24P, V_42P, V_13P, V_31P, V_24N, V_42N, V_13N, and V_31N determine the sample type (n or p) and the sheet carrier density n_s. The Hall mobility can be determined from the sheet density n_s and the sheet resistance R_S obtained in the resistivity measurement. See Eq. (2).

This sequence of measurements is redundant in that for a uniform sample the average Hall voltage from each of the two diagonal sets of contacts should be the same.

Hall Calculations

Steps for the calculation of carrier density and Hall mobility are:

Calculate the following (be careful to maintain the signs of measured voltages to correct for the offset voltage):

	V_C = V_24P - V_24N, V_D = V_42P - V_42N,
	V_E = V_13P - V_13N, and V_F = V_31P - V_31N.		(11)

The sample type is determined from the polarity of the voltage sum V_C + V_D + V_E + V_F. If this sum is positive (negative), the sample is p-type (n-type).

The sheet carrier density (in units of cm^-2) is calculated from

	p_s = 8 x 10^-8 IB/[q(V_C + V_D + V_E + V_F)]
	if the voltage sum is positive, or
			(12)
	n_s = \|8 x 10^-8 IB/[q(V_C + V_D + V_E + V_F)]\|
	if the voltage sum is negative,

where B is the magnetic field in gauss (G) and I is the dc current in amperes (A).

The bulk carrier density (in units of cm^-3) can be determined as follows if the conducting layer thickness d of the sample is known:

	n = n_s/d
	p = p_s/d		(13)

The Hall mobility µ = 1/qn_sR_S (in units of cm²V^-1s^-1) is calculated from the sheet carrier density n_s (or p_s) and the sheet resistance R_S. See Eq. (2).

The procedure for this sample is now complete. The final printout might contain (Sample Hall Worksheet):

Sample identification, such as ingot number, wafer number, sample geometry, sample temperature, thickness, data, and operator
Values of sample current I and magnetic field B
Calculated value of sheet resistance R_S, and resistivity ρ if thickness d is known
Calculated value of sheet carrier density n_s or p_s, and the bulk-carrier density n or p if d is known
Calculated value of Hall mobility µ

Sample Hall Worksheet

Worksheet with Typical Data

IV. Algorithm Example

The sheet resistance R_S can be obtained from the two measured characteristic resistances R_A and R_B by numerically solving the van der Pauw equation [Eq. (3) in the text] by iteration

exp(-p R_A/R_S) + exp(-p R_B/R_S) = 1

as outlined in the following routine:

Algorithm Example

V. References

"Standard Test Methods for Measuring Resistivity and Hall Coefficient and Determining Hall Mobility in Single-Crystal Semiconductors," ASTM Designation F76, Annual Book of ASTM Standards, Vol. 10.05 (2000).
E. H. Hall, "On a New Action of the Magnet on Electrical Current," Amer. J. Math. 2, 287-292 (1879).
L. J. van der Pauw, "A Method of Measuring Specific Resistivity and Hall Effect of Discs of Arbitrary Shapes," Philips Res. Repts. 13, 1-9 (1958).
L. J. van der Pauw, "A Method of Measuring the Resistivity and Hall Coefficient on Lamellae of Arbitrary Shape," Philips Tech. Rev. 20, 220-224 (1958).
E. H. Putley, The Hall Effect and Related Phenomena, Butterworths, London (1960).
D. C. Look, Electrical Characterization of GaAs Materials and Devices, John Wiley & Sons, Chichester (1989).
D. K. Schroder, Semiconductor Material and Device Characterization, 2nd Edition, John Wiley & Sons, New York (1998).
R. Chwang, B. J. Smith and C. R. Crowell, "Contact Size Effects on the van der Pauw Method for Resistivity and Hall Coefficient Measurement," Solid-State Electronics 17, 1217-1227 (1974).
D. L. Rode, C. M. Wolfe and G. E. Stillman, "Magnetic-Field Dependence of the Hall Factor for Isotropic Media," J. Appl. Phys. 54, 10-13 (1983).
D. L. Rode, "Low-Field Electron Transport," Semiconductors & Semimetals 10, 1-89 (1975).