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Hall Effect Measurements
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
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.
Figure 1
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
VH 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 (ns = nd) instead of bulk density. One then obtains
the equation
ns = IB/q|VH|.
(1)
Thus, by measuring the Hall voltage VH and from the known values of
I, B, and q, one can determine the sheet density ns 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 RS 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 = |VH|/RSIB = 1/(qnSRS).
(2)
If the conducting layer thickness d is known, one can determine the bulk resistivity
(ρ = RSd) and the bulk density (n = nS/d).
Figure 2
The van der Pauw Technique
In order to determine both the mobility µ and the sheet density ns,
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
RS. Van der Pauw demonstrated that there are actually two characteristic
resistances RA and RB, associated with the corresponding terminals
shown in Fig. 2. RA and RB are related to the sheet resistance
RS through the van der Pauw equation
exp(-πRA/RS)
+ exp(-πRB/RS)
= 1
(3)
which can be solved numerically for RS.
The bulk electrical resistivity r can be calculated using
r = RSd.
(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. RA and RB are calculated by means of the following
expressions:
RA = V43/I12 and RB = V14/I23.
(5)
Figure 3
The objective of the Hall measurement in the van der Pauw technique is to determine
the sheet carrier density ns by measuring the Hall voltage VH. 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 VH, a current I
is forced through the opposing pair of contacts 1 and 3 and the Hall voltage VH
(= V24) is measured across the remaining pair of contacts 2 and 4. Once the Hall
voltage VH is acquired, the sheet carrier density ns can be calculated via ns
= IB/q|VH| 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
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.
Figure 4
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:
R21,34 = V34/I21, R12,43 = V43/I12,
R32,41 = V41/I32, R23,14 = V14/I23,
(7)
R43,12 = V12/I43, R34,21 = V21/I34,
R14,23 = V23/I14, R41,32 = 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:
R21,34 = R12,43
R43,12 = R34,21
R32,41 = R23,14
R14,23 = R41,32
(8)
The reciprocity theorem requires that:
R21,34 + R12,43 = R43,12 + R34,21,
and
R32,41 + R23,14 = R14,23 + R41,32
(9)
If any of the above fail to be true within 5 % (preferably 3 %), investigate
the sources of error.
Resistivity Calculations
The sheet resistance RS can be determined from the two characteristic resistances
RA = (R21,34 + R12,43 + R43,12 +
R34,21)/4 and
RB = (R32,41 + R23,14 + R14,23 +
R41,32)/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 = RS d can be calculated from RS.
Definitions for Hall Measurements
The Hall measurement, carried out in the presence of a magnetic field, yields
the sheet carrier density ns 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.
V24P = Hall voltage measured between leads 2 and 4 with magnetic field
positive for I13. Likewise for V42P, V13P, and
V31P.
Similar definitions for V24N, V42N, V13N, V31N
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 V24P
Apply a current I31 to leads 3 and 1 and measure V42P
Likewise, measure V13P and V31P with I42 and
I24, respectively
Reverse the magnetic field (negative B)
Likewise, measure V24N, V42N, V13N, and V31N
with I13, I31, I42, and I24, respectively
The above eight measurements of Hall voltages V24P, V42P,
V13P, V31P, V24N, V42N, V13N,
and V31N determine the sample type (n or p) and the sheet carrier density
ns. The Hall mobility can be determined from the sheet density ns and the sheet
resistance RS 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):
VC = V24P - V24N,
VD = V42P - V42N,
VE = V13P - V13N, and
VF = V31P - V31N.
(11)
The sample type is determined from the polarity of the voltage sum VC
+ VD + VE + VF. 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
ps = 8 x 10-8 IB/[q(VC + VD
+ VE + VF)]
if the voltage sum is positive, or
(12)
ns = |8 x 10-8 IB/[q(VC + VD
+ VE + VF)]|
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 = ns/d
p = ps/d
(13)
The Hall mobility µ = 1/qnsRS (in units of cm2V-1s-1)
is calculated from the sheet carrier density ns (or ps) and
the sheet resistance RS. 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 RS, and resistivity ρ if thickness
d is known
- Calculated value of sheet carrier density ns or ps, and
the bulk-carrier density n or p if d is known
- Calculated value of Hall mobility µ
IV. Algorithm Example
The sheet resistance RS can be obtained from the two measured characteristic
resistances RA and RB by numerically solving the van der Pauw
equation [Eq. (3) in the text] by iteration
exp(-πRA/RS)
+ exp(-πRB/RS)
= 1
as outlined in the following routine:
V. References
1. "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).
2. E. H. Hall, "On a New Action of the Magnet on Electrical Current," Amer.
J. Math. 2, 287-292 (1879).
3. 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).
4. 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).
5. E. H. Putley, The Hall Effect and Related Phenomena, Butterworths, London
(1960).
6. D. C. Look, Electrical Characterization of GaAs Materials and Devices,
John Wiley & Sons, Chichester (1989).
7. D. K. Schroder, Semiconductor Material and Device Characterization,
2nd Edition, John Wiley & Sons,
New York (1998).
8. 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).
9. 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).
10. D. L. Rode, "Low-Field Electron Transport," Semiconductors & Semimetals
10, 1-89 (1975).
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