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Transport measurements are necessary in order to monitor the electron sheet concen-tration, mobility and sheet resistance of the two dimensional electron gas, 2DEG. The simplest method to control the sheet resistance, is using the van der Pauw technique which is most accurate for square structures but can also be used for samples of arbitrary shapes. For more detailed analysis Hall effect measurements allow for controlling the additional values of the electron sheet concentration and mobility in the 2DEG. Both of these techniques will be described in this section.

3.1.1 Hall Measurements

The basic principle standing behind Hall measurements is observing the Hall voltage produced when electrons flowing from one end of the channel to the other are slightly deviated in the direction perpendicular to their flow with the application of a magnetic field. The phenomenon that causes an electron to move in a direction that is perpendicular to its direction of motion and the applied magnetic field is is known as the Lorentz force, see figure 3.1. From this force it is possible to measure the Hall voltage, V_H. Then with the known values for the current, I, the magnetic field, B, and the electron charge, q=1.602x10⁻¹⁹C it is possible to calculate the electron sheet concentration, ns, equation 3.1. The Hall effect is usually used to measure a three dimensional channel, however since we are concerned with the two dimensional 2DEG, all of the equations listed in this section will be for two dimensional channels.

n_s = IB

q|V_H| (3.1)

If the sheet resistance R_s of the semiconductor is known it is then possible to calculate the carrier mobility, µ, with equation 3.2.

µ = |V_H|

R_SIB = 1

qn_sR_s (3.2)

36 CHAPTER 3. MEASUREMENT TECHNIQUES

Figure 3.1: Hall Measurements. Image is taken from http://www.eeel.nist.gov/812/effe.htm.

The sheet resistance, R_s, for a Hall bar structure is calculated by equation 3.3, where I_c is the current applied to the extremities, W the width of the Hall bar, L the length of the Hall bar and V the voltage drop across the Hall bar. Equation 3.3 is a two dimensional measurement and is valid for the resistance describing the two dimensional electron gas.

If it is necessary to measure R_s for a sample without the Hall bar patterned onto it or a arbitrary shape sample one must follow the method described by Pauw [1958/59].

R_s = V W

I_cL (3.3)

3.1.2 The van der Pauw Technique

For this thesis the van der Pauw technique was useful in preliminary measurements of de-terming the sheet resistance of the 2DEG and observing its change during the ferroelectric deposition process. This measurement technique was less cumbersome than preparing structure samples for Hall effect measurements. Using the van der Pauw technique the sheet resistance Rs can be calculated easily for arbitrary shaped samples, equation 3.5, Pauw [1958/59]. This is done by applying a voltage across two neighbouring electrodes and measuring current across the other two neighbouring electrodes, see figure 3.2 where 1, 2, 3, and 4 denote the four corner electrodes. This is done in two different orientations to allow for the calculation of R_A and R_B using the equations 3.4.

R_A = V₄₃

I₁₂ R_B= V₁₄

I₂₃ (3.4)

e^−πRARS + e^−πRBRS = 1 (3.5)

If wanting to do more accurate measurements for R_A and R_B one needs to take into account the a-symmetrical effects produced by the sample to do so more current-voltage needs to be done.

3.1. TRANSPORT MEASUREMENTS 37

Figure 3.2: Van der Pauw measurements. Image is taken from http://www.eeel.nist.gov/812/effe.htm.

R_A = (V₃₄

Where due to symmetry the following equations 3.8 - 3.10, must be true within a 5%

margin of error.

And due to reciprocity the following equations must hold true, equations 3.9 and 3.10.

V₄₁

Using the van der Pauw resistivity measurements the carrier mobility and the carrier concentration can also be measured. As seen in figure 3.3, the current is applied to two terminals opposite from each other and the voltage is measured from the other two terminals. This voltage labeled V_24P is equivalent to the Hall voltage as measured in the Hall measurements. Afterwards the same equations as for Hall measurements are used to calculate the carrier mobility and concentration. These techniques can allow for the studying of the 2DEG properties in the ferroelectric/heterostructure device as a function of ferroelectric polarisation, however n_s and µ are more accurate when using a Hall bar structure.

38 CHAPTER 3. MEASUREMENT TECHNIQUES

In fact when measuring the Hall effect for an arbitrary shape it is also important to check for symmetry. The most accurate results will thus be obtained by taking measurements when the magnetic field is applied in both polarities and reversing the current/voltage configuration. In this manner one will have an averaged value for the Hall voltage. This is explicitly done by applying a positive magnetic field and then: apply I₁₃ and measure V_24P, apply I₃₁ and measure V_42P, apply I₂₄ and measure V_13P and apply I₄₂and measure V_31P. Then the same needs to be done for a negative magnetic field: apply I₁₃ and measure V_24N, apply I₃₁ and measure V_42N, apply I₂₄ and measure V_13N and apply I₄₂and measure V_31N. Then it is possible to calculate V_C, V_D, V_E and V_F, see equation 3.11, to calculate the Hall voltage of the sample with equation 3.12. It is then possible with the known values for V_H and R_s to calculate the electron sheet concentration with equation 3.1 and the mobility with equation 3.2.

V_C = V_24P−V_24N V_D = V_42P−V_42N V_E = V_13P−V_31N V_F = V_31P−V_31N (3.11)

V_H = V_C+ V_D + V_E+ V_F

8 (3.12)

Figure 3.3: Measuring the Hall effect by the van der Pauw technique. Image is taken from the website http://www.eeel.nist.gov/812/effe.htm.