RESISTIVITY OF A SEMICONDUCTOR BY THE FOUR-PROBE METHOD

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1 1 Experiment-322 A RESISTIVITY OF A SEMICONDUCTOR BY THE FOUR-PROBE METHOD Dr Jeethendra Kumar P K, Tata Nagar, Bengaluru INDIA. Abstract Resistivity of an n-type germanium semiconductor sample is determined by using a four-probe set-up. A constant current is passed through the two outer probes and the voltages developed across the two inner probes are recorded. Knowing the thickness and width of the semiconductor sample, finite thickness (f 1 ) and finite width (f 2 ) corrections are applied and the resistivity of the sample is calculated. Introduction The electrical resistance R of a material is defined as the ratio of voltage applied across it to the current that causes it R = 1 For a metallic wire the resistance is given by R = 2 where ρ is the resistivity L is length of the wire, and A is cross sectional area of the wire. Resistivity ρ = 3 Hence in resistivity measurements the dimensions of the material (length and cross sectional area of the wire in this case) are also required. Based on the electrical resistivity, ρ, materials are classified as (i) conductors (metals), (ii) semiconductors, and (iii) insulators. Resistivity is the opposition or resistance offered by a unit cube of the material at its opposite faces to the flow of electric current passing through it. Resistivity is measured in Ohm-meter or Ohm-cm (earlier measured in Ohm. square). The reciprocal of resistivity is conductivity (σ), which is measured in Ohm -1 meter -1 (Mho/meter). The conductivity of a semiconductor increases with increase in temperature (i.e., negative temperature coefficient of resistance). Metals and insulators have opposite behavior (i.e., positive temperature coefficient of resistance). At absolute zero temperature, a semiconductor

2 2 also acts like a pure insulator. There are two types of semiconductors, namely (i) Intrinsic semiconductors (pure or natural semiconductors), and (ii) Extrinsic semiconductors (impure or artificial semiconductors) [1]. For number of years the resistivity, surface resistivity, sheet resistivity or bulk resistivity were measured just as a number without mentioning their dimensions [2]. Valdes in 1954 first reported the measurement of germanium resistivity using four probes [3]. Later Uhlir (1955) assumed three dimensional structures with one infinite dimension [4]. Smits (1958) extended this work for a two-dimensional structure. He defined a four-point probe method of measuring sheet resistivity [5]. This work eventually became an industry standard for measuring the resistivity of semiconductors. He developed correction factors for measuring sheet resistivity on two-dimensional and circular samples. He found that this method was not only useful for measuring resistivity of diffused surface layers, but also useful in obtaining body resistivity of thin samples (thin films). Resistivity of semiconductor materials Electrical resistivities of semiconductor materials are very important for making electronic devices; the purity of the material being the most important quality for this purpose. Hence to account for the purity, resistivity is taken as the parameter. The purity of a Germanium semiconductor was checked by measuring its resistivity along the length of the sheet at closely spaced intervals using the four-probe setup. Greater the impurity, lower is the resistivity. Hence resistivity is being considered as the most important parameter in semiconductor industry. Figure-1 gives a list of semiconductors and their resistivity values. Pure germanium has a resistivity of 60 Ohm.cm. Pure silicon has a considerably higher resistivity, of the order of 60,000 Ohm-cm. By doping (i.e., adding an impurity) a semiconductor, conductivity changes considerably. Figure-1: Range of resistivity values of a metal, a semiconductor and an insulator The four-probe method A digital multimeter (consisting of two probes) can measure resistance accurately above 1Ω. To measure resistance below 1Ω it is not a good option because the lead wire (probe) used to measure resistance itself has probe resistance (R p ) of the order a fraction of an Ohm that comes in series with the resistance to be measured. There is an added problem with the measurement of small resistances using two probes, especially in the case of semiconductors. The probe used to measure resistance consists of a metal. When it is placed over a semiconductor, the metal-semiconductor interface acts like a rectifier (point contact diode). Hence the resistance measured is directional. The contact

3 Lab Experiments 3 resistance between the metal-semiconductor interfaces comes in to play. The current (electron flow) from the probe flows to the semiconductor sample and it spreads in all directions as shown in Figure-2. The spreading of charges under the probe head also contributes to resistance, called spread resistance (R sp ). Hence in a sheet of material, for resistance measurement the following need to be considered: 1 The probe has its own resistance called probe resistance (R p =ρl/a). 2 At the interface between the probe tip and the semiconductor, there is probe contact resistance (Rcp). c 3 When the current flows from the tip into the semiconductor and spreads out in the semiconductor, there will be a spreading resistance (R sp ). 4 Finally, the semiconductor itself has a sheet resistance (R s ). One needs to account for these resistances in order to get an accurate value of resistivity of a material [6]. Using four probes instead of two, one can eliminate the probe resistance, contact resistance and spread resistance in the measurement. Figure-2: Various resistances associated with a semiconductor sample and measuring probes Figure-3 shows the equivalent circuit of four probe measurements. A voltmeter with very high input resistance ( 1 MΩ) is used to measure voltage across the middle two probes. Therefore a very small amount of current flows through probes 2 and 3. Further, the current flowing in opposite directions in probes 2 and 3 cancel out each other. The role of the probes can be interchanged, i.e. current can be passed through the inner probes and voltage can be measured across the outer probes. Now applying Kirchhoff s voltage law to the closed loop associated with the probes 2 and 3 (middle two probes), one obtains; V = R s (I-I p ) + I p (R p2 +R cp2 +R sp2 ) I p (R p3 +R cp2 +R sp2 ) 4 Since the four probes are identical in dimension and parameters, this becomes R p1 =R p2 =R p3 =R p4 ; R cp1 =R cp2 =R cp p3=r cp4 and R sp1 =R sp2 =R sp3 V = R s (I-I p ) 5

4 4 Since I P is very small (about 10 6 times smaller than I) V= R s I 6 Hence the voltmeter measures only the sheet resistance of the sample. The probe resistance, contact resistance and spread resistance cancel out in this method of measurement. Hence it is an accurate method of measuring the sheet resistance. I V I Rp1 Rp2 Rp3 Rp4 Rcp1 Rcp2 Ip Ip Rcp3 Rcp4 Rsp1 Rsp2 I-Ip Rsp3 Rsp4 Probe-1 Probe-2 Probe-3 Probe-4 Figure-3: Various resistances coming into play in the four- probe method of measurement; the equivalent circuit of four probes Boundary conditions in the four-probe resistivity measurement Experimentally it is observed that the resistivity measured using the probe varies with the geometry (size) of the sample, its thickness, its shape and placement of the probes on the surface. The resistivity values measured by placing the probe at the middle of the sample and that on one side of the sample were different. It also varies with the distance between the probes or probe gap. Alignment of the probe in line or off-line is also found to yield different resistivity values. Further, it also depends on the nature (smooth or rough) of the surface over which the material is placed. Hence resistivity measurement with four probes is complicated. Further, it involves considerable amount of mathematical calculations. To account for these variations; boundary conditions are introduced. These boundary conditions are nothing but correction factors for various parameters contributing to the resistivity measurements [5]. After vigorous experimentation at Bell Labs on germanium semiconductor, A Uhlir Jr and F M Smits put forward various boundary conditions and accounted for different aspects of resistivity measurements. They presented a correction factor in the form of a graph, depending on the size (area) and shape (circular, rectangular and irregular) of the material, its thickness in relation to the probe gaps and surface nature (conducting or non-conducting bottom surface) on which it is placed. In these calculations the gap between the probes is held constant and the probes are aligned in a line and placed at the center of the material during the measurement.

5 Lab Experiments 5 In practice the gap between the probes is taken as 2mm and all the four probes are aligned in a straight line as shown in Figure-4(a). Figure-4(b) shows unequal gaps between the four probes, uneven rough surface sample and different probe heights. Each of these conditions has different boundary conditions. In this experiment we have used a 2mm x 8mm rectangular sample of 0.5mm thickness. Hence the subsequent discussion is confined to our sample only. Figure-4: (a) Equal probe gaps, plane sample, equal probe height (b) Uneven sample surface, unequal probe gaps, with probes aligned at different heights Resistivity of a semi-infinite volume sample A semi-infinite sample is one that extends to infinity in all directions below the plane, on which the four probes are placed. Practically it is a thick and large lamina. The four-probe is placed at the centre of this sample and a constant current is passed through the outer probes. The floating potential developed across the middle probes is given by V f = 7 where ρ o is the resistivity of the semi-infinite semiconductor material; I is the current passing through the outer probes; and r is the distance from the current carrying probe to voltage measuring probe. Probes 1 and 4 carry current hence the floating potential under probe 2, due to flow of current in Probe-1 and Probe-4 is given by V f2 = 8 The floating voltage produced under Probe 3 due to the flow of current in Probe-1 and Probe- 4 is given by V f3 = Hence the voltage measured across the inner two probes is nothing but difference across Probes 2 and 3 9 the potential

6 6 V= V f2 -V f3 = - Since we have used a probe with equidistance or gap of 2mm, we have S 12 =S, S 23 =S, S 34 =S. Hence V = V = + V = V = + This is the potential measured across the inner probes of the four probe set-up, from which resistivity is given by ρ = 2πS 10 where I is current passing through the outer probes V is the voltage developed across the inner probes S is distance between the probes ρ 0 is the resistivity of semi-infinite sample 2πS is known as the geometric factor. This is the value of resistivity of a semiconductor measured using very thick and large sized or a semi-infinite sample. Resistivity of a finite-sized sample Figure-5: Samples with finite dimensions A finite sample has thickness t and width d and length in multiples of d, as shown in Figure- 5. It can be a square, rectangle or circular sample with fixed thickness. The thickness (t) and width (d) are compared with the probe spacing (S) in all measurements. The measurements are made with respect to the ratios t/s and d/s. Hence these two ratios are very important in the resistivity measurements.

7 7 It is observed by experimentation that the sample with thickness-gap ratio t/s 0.5, the geometric factor 2πS is given by 2πS = =4.53t for a square, rectangle sample We have used a sample of t=0.5mm and S=2mm. The ratio t/s = 0.25 which is <0.5. Further, the width d=8mm or the ratio d/s= 8/2=4. Thus d>s. Hence the resistivity of sample having thickness t<<s and width d>>s, with the four probes placed at the center of the sample, is given by [5] ρ = 4.53t ( ) 11 For a sample of finite width and non-negligible thickness this should be multiplied by f 1 and f 2, the two correction factors. Hence resistivity is given by ρ = 4.53t ( )f 1f 2 12 where f 1 is the finite thickness correction factor, and f 2 is the finite width correction factor. The finite thickness factor varies with the ratio t/s and depends on the shape of the sample (circular or rectangular), whereas the finite width correction factor f 2 varies with the ratio d/s. Figure-6 and Figure-7 show these variations as calculated by Smith. One can determine the values of f 1 and f 2 from these graphs and substitute them in Equation-12 to determine ρ. The values of f 1 and f 2 are calculated by Smith and Uhlir. The finite thickness correction for an insulating bottom boundary is given by f 1 = f 11 ( ) = ( ) ( ) for an insulating bottom boundary 13 We have pasted a mica sheet to the bottom of the sample, hence in our case f 1 =f 11 (Figure-6). For a conducting bottom surface f 1 = f 12 ( ) = ( ) ( ) for an conducting bottom surface 14 In the f 2 graph shown in Figure-7, five different closely related curves are given by Smith. One can choose the curve by knowing the dimension of the sample. Before performing the experiments the thickness (t) and width (d) of the sample are noted first. The ratios t/s and d/s are calculated. Corresponding to these ratios the value of f 1 and f 2 are noted from Figure- 6 and Figure-7 respectively. Semiconductor samples

8 Lab Experiments 8 Semiconductor samples used in these experiments are obtained from the scrape material generated by the semiconductor industry. These materials are obtained in varying shapes and sizes. We have cut them into pieces of about 12mm x 5mm size. However, due to manual cutting it is difficult to get exact size; hence there will be a slight difference in the dimensions of different samples, especially the width of the sample which appears into calculation. Hence the width of the sample has been measured using a travelling microscope at different positions of the sample and an average value is taken as its width. Figure-6: Finite thickness correction for resistivity measurement by the four-probe method..

9 9 Figure-7: Finite width correction for resistivity measurement using four probes Apparatus used The experimental set-up for measurement of resistivity of a semiconductor by the four- probe method consists of: digital milli-volt meter, digital milli-ammeter and constant current source and n-type semiconductor sample. Figure-8 shows the complete experimental set-up.

10 10 Experimental procedure Figure-8: The Four-probe experimental setup 1. We have used three samples of different widths in this experiment to determine the resistivity. The circuit connections are done as shown in Figure-9. The four probes are connected to their respective sockets on the experimental set-up. DCM + - Constant Current Source ma + - mv DVM Set- Current 0.5mm 2mm Semiconductor sample Figure-9: Four probe circuit connections 2. The thickness (t), width (d) and gap between the probes (S) are noted for the experimental set-up as Gap between the probes (S) =2mm

11 11 Thickness of the sample (t) = 0.5mm These parameters are the same all the three probes. Only their widths are different, as shown in Table-1. Table-1 Sample Width d(mm) f 1 f Sample width and correction factors 3. For the first sample the ratio is obtained as = = Since the bottom surface is non-conducting f 11 corresponding to t/s =0.25 is noted f 1 =f 11 from graph 1 is given by f 1 =1 Similarly the ratio =. = 2.7 The value of f 2 corresponding d/s = 2.7 is noted from the Figure-7. f 2 = 0.59 The correction factors for the three samples are listed in Table The sample is placed under the four probes such that the probes are parallel to the longer side (length) of the sample. Figure-10 shows the sample and four probe arrangement. Figure-10: Germanium sample (5.4mmX12mm) and the spring load four probe arrangements

12 12 5. The 4mm banana pins coming out of the two outer probes are connected to a constant current source in series with a digital milli-ammeter. The inner probes with 2mm banana pins are connected to the milli-voltmeter. 6. The current is set to 0.1mA by adjusting set-current knob. The voltage developed across the inner probes is noted I =0.1mA and V= 5.4mV Hence V/I = 5.4mV/0.1 = The trial is repeated by varying the current in steps of 0.1mA up to a maximum of 1.5mA. In each case the voltage developed across the inner probes is noted and V/I ratio is calculated. The readings obtained are tabulated in Table-2 and the average value of V/I is calculated. Current (ma) Table-2 Sample-1 Sample-2 Sample-3 mv V/I mv V/I mv V/I Average V/I Current to voltage ratio for three different samples 8. Resistivity is calculated for sample using Equation-12 as ρ = 4.53t ( )f 1f 2 = 4.53 x0.5x10-3 (53.0) x1x 0.59= Ωm = 7.08Ωcm 9. The experiment is repeated for sample-2 and sample-3. The current passed and voltages developed are tabulated in Table-2 and their resistivities are calculated. For sample-2 ρ = 4.53t ( )f 1f 2 = 4.53 x0.5x10-3 (60.2) x1x 0.54= Ωm = 7.36Ωcm

13 13 For sample-3 ρ = 4.53t ( )f 1f 2 = 4.53 x0.5x10-3 (73.5) x1x 0.46= 0.076Ωm = 7.6Ωcm The average value is taken as ρ of the given germanium semiconductor ρ = 7.34 Ωcm Graphical method Instead of taking average value of V/I, one can also draw a graph of V and I as shown in Figure-11 and its slope can be taken as the average value of V/I sample-1 sample-2 sample-3 Voltage (mv) Current (ma) Figure-11: The V/I graph Slope of the graph for different samples is calculated from the three straight lines and substituting it in equation-12 one obtains resistivity as Sample-1 slope =V/I =52.5; which gives ρ = 7.0 Ωcm Sample-2 slope =V/I =59.0; which gives ρ = 7.1 Ωcm Sample-3 slope =V/I =72.3; which gives ρ = 7.4 Ωcm Results Resistivity of n-type Germanium semiconductor = 7.34 Ωcm (obtained from the calculations) Resistivity of n-type Germanium semiconductor = 7.16 Ωcm (obtained from the graphical procedure) References [1] DC Tayal, Electricity and Electronics, Himalaya Publishing House, 1983, Page-674.

14 14 [2] Gene Chase, [3] Valdes L, Resistivity measurements on germanium for transistors, Proc. of IRE, 42, b, 1954, P420 [4] Uhlir A Jr, The potentials of infinite systems of sources and numerical solutions of problems in semiconductor engineering, B S T J 34, Jan, 1955, Page-105. [5] F M Smits, Measurement of sheet resistivities with the four-point probe, the Bell system technical journal, May 1958, Page-711. [6] K X Chen, J K Kim, F Mont, and E F Schubert, Four-point probe measurement of semiconductor sheet resistance.

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