International Journal of Natural and Applied Sciences, 7(4): 364 – 369, 2011 ISSN: 0794 – 4713

www.tapasinstitute.org/journals/ijonas ©Tapas Institute of Scientific Research and Development, 2011

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ELECTRO OPTIC HALL PROBE

A. Yakubu Department of Physics, Kebbi State University of Science and Technology, Aliero, Nigeria Adrian

Boland Thons, Department of Computer and Electronic Engineering, University of Essex, United Kingdom

E-mail: abulect73@yahoo.com

ABSTRACT

Electro optic voltage probing is a non-invasive technique for determining voltages in crystals which

exhibits induced birefringence. This technique allows the transport properties of semiconductors to be determined without the physical limitations of standard characterization techniques. In this

experiment, the Hall voltage was measured for bulk Silicon doped Gallium Arsenide, thus mobility,

μ, of the carriers was also obtained. The theory of this technique was given, showing the analysis of the Hall voltage. Amongst others, we found some anomalies in the experiment which could be

explained by the assumption that a floating voltage exists on the non-metalized surfaces of the sample. This was independent of the Hall voltage but not the applied voltage.

Keywords: Electro optic, Hall voltage, semiconductors, Gallium Arsenide

INTRODUCTION

A large amount of theoretical and experimental work has been undertaken on Electro optic probe.

A novel method of determining the mobility of carriers in Gallium Arsenide and other electro optic crystals was first proposed by Andreou and Christodoulides (Lyons, 1992). The electro optic Hall

probe combines the standard Hall Effect method for measuring mobility and the electro optic method for measuring voltages in certain crystals. This experimental research verifies this theory in

bulk GaAs. Carrier mobility is the most fundamental transport property of a semi-conductor. From available

technology today, mobility are in excess of 2 x 106 cm2/vs(Willardson). The carrier properties of a

semi-conductor, such as carrier type, concentration and mobility can be determined using Hall Effect. The basic principle of Hall Effect is the movement of carriers, induced by the Lorentz force.

Electric and magnetic fields are applied perpendicularly to the semi-conductor which forces the carriers to move in the direction of both the electric field, Ex and the magnetic field By.

The movement of these carriers set up a potential difference in the same direction. The potential

difference set up is known as the Hall voltage. The Hall voltage denoted by VH, is given as;

........................................................................1.1

Where Lz is thickness of sample in Hall voltage direction, Lx is the length along which Vx is applied

and Vx is the applied voltage. The Hall factor is dependent on the scattering mechanism of the carriers.

There are two main standard methods for determining the mobility of carriers; these methods are the Hall bar configuration and the Van Der Pauw configuration (Tyagi).

Hall bar configuration: It’s very important that the probes are very tiny; having a high resistance

so they can not draw current from the main bar. The tiny nature of the probes also prevents any voltage from developing across their width. Probes position must be aligned correctly to avoid

partial measurement of the applied potential difference. Van Der Pauw configuration: This is simpler configuration where mobility is measured across

different combinations of contacts to give an average over the sample. The Van Der Pauw consists

of two parts. The first part gives a value for resistivity, ρ, of the sample with no magnetic field. In the second part, the Hall factor, RH is obtained from the formula;

………………………………………………..1.2

RH = 1/ne …………………………………………….1.3

………………………………………………..1.4

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The relationship in 1.4 is true for a quasi classical probe of the Hall Effect. Both of the methods

discussed have their limitations when measuring the mobility of carriers. The thin probe in the Hall bar are difficult to fabricate coupled with the addition of a metalized probes in both cases.

These problems may alter the charge distribution of the sample. Fontein et al. (1992) showed in

their experiment the nature of this alteration using electro optic probing. For high speed transient signals, reflection of the electrical signal will occur if the circuitry is not impedance matched. The

Hall bar is very difficult to impedance match in this regard. Hence the Hall bar is not suitable for high speed transient voltage measurement (Yakubu, 2009).

The electro optic voltage probe uses Pockels effects which occur in non-centrosymmetric crystals (Yakubu, 2010). These crystals which include Gallium Arsenide have no spatial inversion symmetry

about any point in their lattice. When a light beam passes through a non-centrosymmetric crystal

whilst an electric field is applied parallel to the light, the polarization of the beam is shifted. The magnitude of this shift is directly proportional to the magnitude of the applied filed. If the shift in

polarization is measured, the strength of the applied field could be obtained. This method of voltage probing is non-invasive. It only requires the electrical contacts necessary to

apply the voltage to the sample, and no probing contacts are required.

MATERIALS AND METHODS

Theory: The theory of electro optic probing and how it is applied to the Hall Effect in addition to

the theory of Pockels Effect were used to show that the change in phase of a beam of light is directly proportional to the magnitude of an electric field applied to any material through which a

the beam passes. This theory is applied to the Hall Effect to show Hall voltage, and hence the

carrier mobility may be measured. Pockels Effect and electro optic voltage probe: Pockels Effect occurs in crystals with no

centre of inversion (non-centrosymmetric). GaAs is non-centrosymmetric crystal used in our experiment. When electric field is applied across such a crystal, birefringence is induced. The

alteration in the refractive indices of the crystal in three crystallographic directions by the

application of the electric field causes birefringence (Yakubu, 2010). This alteration causes a change in the velocity of the two components of polarization of a light beam propagating through

the crystal. This change in velocity also causes a change in phase to the incident beam. To understand the change in phase, the refractive index is described by an index ellipsoid.

The index ellipsoid relates the refractive indices of the crystal along the three principal axes, x, y,

and z, and direction of the applied electric field is chosen to correspond with one of the axes. In our case, the z axis was chosen.

The value of the refractive indices in the directions perpendicular to applied electric field, Ez , i.e, nx, ny, are obtained by considering the intersection of the plane perpendicular to Ez. The index

ellipsoid is given by

…………………………….2.1

Where x, y , z are the crystallographic axes of the crystal, no is the unperturbed refractive index

and E = (Ex, Ey, Ez) are the electric field present in the crystal. rk1 is the electro optic tensor of the crystal.

Polarization of light beam propagating through a crystal has the formula;

Where ϵo is permittivity of free space, x1, x2, x3 are linear, quadratic and cubic susceptibility

respectively. The second and third susceptibility are Pockels and Kerr coefficients respectively.

Since the susceptibility x1>>x2>>x3, the polarization of a beam is directly proportional to electric

field for low fields. Thus for GaAs, the tensor is r41. The Pockels effect starts to occur for GaAs at fields of about 10Kv/m. Perturbation of refractive index due to electric field is;

The value of the perturbed refractive index in the x and y directions are given as;

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For the purpose of the technique used in this experiment, the light is assumed to propagate parallel

to electric field; Ez. from equation 2.4 and 2.5, there is no perturbation of the refractive index if the applied field is zero. When field is applied, the components of polarization no longer propagate with

the same velocity. As a result, there is a shift with respect to one another and the light emerges

elliptically polarized. The magnitude of this shift is directly proportional to the shift in refractive index which is also proportional to the magnitude of the applied electric field.

The magnitude of the shift (retardation), Г, is given as

Where, Фx and Фy are the perturbed polarization components in the crystallographic directions.

The phase’s Фx and Фy are proportional to the refractive indices and the length of the crystal, Lz, through which the beam passes, as shown in the formula below;

This allows the retardation to be described in terms of the perturbed refractive indices;

From equation 2.4 and 2.5, the retardation can be related to the applied electric field by;

The half wave voltage is given as,

Thus the shift in polarization of the beam may be written as;

Hence;

Where Vz is the voltage applied to produce the field Ez. This mathematically proof has shown that the retardation of the polarization of a light beam

propagating through a crystal which has an electric field applied to it in the direction of propagation is directly proportional to the magnitude of the electric filed. Thus applied voltage may be found by

measuring the shift in polarization of a beam of light passing through a crystal.

The Hall Effect: The Hall Effect is the most commonly used methods for determining the concentration and mobility of carriers in semi-conductors. It is also used to indicate whether the

carriers are holes or electrons. The Hall Effect occurs when electric and magnetic fields are applied, perpendicularly to one

another, across a semi-conductor. The applied fields set up a force on the carriers in the same manner as the force in wire placed in orthogonal electric and magnetic fields.

From Lorentz equation, the magnitude of the field is given by;

F = e(E + VxB) = 0………………………………………..2.13 Where e is the charge on the carriers in the crystal, B, is the applied magnetic field and V is the

velocity of the carriers. This equation can be reduced to; eEz = -eVxBy……………………………………………2.14

Current, Ix is produced as a result of the applied field Ex and By and Vx. The current Ix is given as;

Ix = AVxne……………….………………………….2.15 A is cross sectional area equal to LyLx and n is carrier concentration of the semiconductor, hence

substituting for Vx in equation 2.14 gives;

By definition, Ez is the hall field EH. This means that the Hall voltage, VH = EHLz, where Lz is the

thickness across which the Hall voltage is set up. With further simplification, the Hall voltage is;

VH =

Expressing the Hall voltage in terms of Hall mobility, this will give;

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The Hall mobility could be obtained from equation 2.18 by measuring the Hall voltage. Electro Optic Hall probe in bulk semiconductors: The combination of the two theories

discussed is combined to give a new method for measuring carrier mobility. To relate the Hall voltage to the measured voltage, the transmission through the sample must be considered. If a

modulated applied electrical signal is used, the Hall voltage, and the polarization shift will also have

a modulated form. Hence the beam will consist of an a.c. modulated components and a d.c. offset, which is due to the intensity of the beam. This is defined as the ratio of the transmitted to the

incident intensity (Vickers 1992). Thus,

The change in transmission with retarding voltage, ΔV, evaluated at the voltage at which the polarization is retarded by π/2 radians, Vπ/2, is given as (Yariv),

Also, the a.c. component of the beam is proportional to the experimentally measured voltage V rms. That is, Vac α Vrms , hence, VacA = Vrms. Where “A” is a function of the measuring system used.

From equation 2.20, the transmitted intensity is given by;

Hence,

In electro optic Hall probe, the modulating voltage is the Hall voltage. Therefore, equation 2.23 will

become;

From equation 2.18, in terms of the Hall mobility, the ratio becomes;

Hence, mobility could be expressed as;

The above equation is used to calculate the Hall mobility from known and measurable quantities.

Experimental test: The sample under study was the bulk Gallium Arsenide. The samples used were taken from wafers provided by EPI Materials Limited, with orientation (100) 40 (110), and

thickness 500 μm. The sample was doped with silicon giving a carrier concentration n = 3.5 x1018

cm-3. The mobility of the wafer was measured by EPI materials to be 1590 cm2/Vs.

In our experiment, using the Van Der Pauw method, the mobility was found to be 1572 cm2/Vs. In

measuring the Hall Voltage, two main methods were used. The first method was by setting up a known voltage using a very thin Gold disc in the probing direction. The retardation due to this

applied voltage was measured. In the second method, sample without the Gold disc was used and to determine the Hall voltage, an analytical method was employed.

After ensuring the right connections, the ohmic behavior of the connection were ensured by measuring the voltage/current characteristics of the sample. Both d.c and a.c voltages were used,

the d.c voltage used was below one volt to avoid joule heating. While a.c voltages were applied

using the signal generator. Readings were recorded for both forward and reverse voltage application.

The Hall probe experiment was carried out by measuring Vrms as a function of applied magnetic field for both samples. The experiments were carried out for different sets of circumstances so as

to check validity of the results obtained and to investigate any anomalies found. The slope of the

graph of Vrms against B (Magnetic field) was used in equation 2.26 to calibrate the amplification factor, A, of the system against the values of the sample mobility obtained from the EPI materials

and the Van Der Pauw experiment.

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RESULTS AND DISCUSSION

The average gradient obtained from the graph of B against Vrms/Vdc was;

(4.95

This value was substituted into equation 2.26 and equated to the value of mobility given by EPI

materials. This allowed for calibration of the system, yielding an amplification factor, A. To calculate

A, the following values were used Vπ = 9.3kV, Lx = 1 x 10-3m, Lz = 0.5 x 10-3m and Vx = 15V. Using the values of the mobility by EPI and the Van Der Pauw, the electro optic Hall Probe measurement

gave the amplification factor of the system to be;

A = 2.85

Results from different samples of same substrate: Experiment carried out on three different samples showed that the gradients of the graph plotted for B/Vrms were not identical. Their

gradients showed variation. Figure I showed that the samples in this case had no Gold disc.

The Background Electro optic Effect for Bz = 0: The graph of V against Vrms/Vdc was plotted to investigate background of Electro optic effect. The result shows a variation in the normalized

measured voltage with applied voltage. The sample investigated was highly doped and presented a far from linear graph (Figure II).

Voltage/Current characteristics of the samples: These characteristics are shown in figure

IIIa and IIIb for both d.c and a.c for reverse and forward bias applications. The graph showed the ohmic nature of the sample to be in agreement with the stated law.

Fig.I: Magnetic field against Vrms/Vdc for different samples with applied voltage of 15V, sample length, Lx = 1mm, sample thickness, Lz = 0.5mm

Fig. II: graph of applied voltage against Vrms/Vdc.

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Fig. IIIa: d.c voltage/current characteristics for sample 1

Fig. IIIb: a.c voltage/current characteristics for sample 1

CONCLUSIONS

There is variation in the normalized measured voltage with applied voltage as shown in figure II. The possible explanation to this anomaly is by assuming that the floating voltage exists on the non

metalized surfaces of the samples. Finally, this experiment succeeded in measuring the Hall Effect and hence the carrier mobility of GaAs using Electro optic Hall probes.

REFERENCES

Fontein, P. F., Hendricks, P. and Ginkel, J. (1992). Linear electro optic effects in zinc-blended type. Surface Science Letter, 61(11): 91 – 96.

Tyagi, M. (1991). Introduction to semiconductor materials and devices. Wiley Publication, Cambridge shire, CB6 3NW.

Vickers, A. J., Tsui, E. S. M., Robert, F. and Lyons, V. (1992). Very high frequency and microwave

interferometric phase and amplitude noise measurement. International Review of Scientific Instruments, 63(11): 5487 – 5488.

Willardson, K. and Beer, T. (1975). Semiconductors and semimetals. Academic Press, New York. Yakubu, A. (2009). Electro optic sampling techniques for THz detection. University of Essex,

Colchester, CO4 3SQ, United Kingdom.

Yakubu, A. (2010), High frequency THz devices. University of Essex, Colchester, CO4 3SQ United Kingdom.

Yariv, A. (1989). Quantum physics. Wiley Publication EPI Materials Ltd, 3 Lancaster Way Business Park, Ely, Cambridge shire, CB6 3NW.