FT09 MW / Page 1 of 9 Introduction : A basic property of a conductive material is its electrical resistivity. The electrical resistivity is determined by the availability of free electrons in the material. In turn the availability of free electrons is determined by the physical binding properties of the material on a molecular level. Other important properties of the material are related to the physical binding properties and therefore to the electrical resistivity of the material. Much may be learned about the properties of a material by measuring its resistivity. An important example is the characterization of semiconductor materials where the resistivity is strongly related to the level of purposely added impurities. Measurement of the resistivity is used to both characterize the material and as a process control parameter for the semiconductor manufacturing process. Resistivity measurements are also used to characterize many other materials such as the magnetic films used in CDs etc. This paper discusses the measurement of electrical resistivity using one of the oldest technologies available, four in line probes of constant spacing contacting the surface of the material. Electrical current is passed between two of the probes and the voltage created is measured between the other two probes. The resistivity may be calculated from the measured values of current and voltage as noted below. The concept of sheet resistance and its relationship to the material s resistivity is discussed before the equations for calculating resistivity are presented. The concept of Sheet Resistance : Length Surface 2 Surface 1 Thickness (t) Width (w) Figure 1 Please examine Figure 1, a thin conducting sheet with current passing through the ends of the sheet from surface 1 to surface 2. The sheet has a length and a cross sectional area (A). The current generates a voltage difference between surfaces 1 and 2. The average resistance, R, between surfaces 1 and 2 may be calculated from Ohms Law as R = V/I (1) Where V is the voltage difference between surface 1 and surface 2 and I is the current. R is also given by R = resistivity x length / A (2)
FT09 MW / Page 2 of 9 Where resistivity is an electrical property of the material in ohm-cm The length of the sheet is measured in cm A is the cross sectional area surfaces 1 and 2 and is equal to t (thickness) x w (width) in cm x cm. Then R = resistivity x length/ (t x w) (3) A useful concept is to visualize the special case of the resistance of one square (length = w) of the thin sheet as viewed from the top. By substitution, the resistance for the square is given by Rs = resistivity / t (4) The result is accurate for one square of conducting material so the units for Rs are given as ohms per square. (Strictly speaking the square has no physical meaning and the units are in ohms.) By substitution the value of Rs may be measured and the resistivity calculated if V and I and t can be measured for the sample. Difference between 2P and 4P : I V I V 2 point probing 4 point probing With the 2 point method, current and voltage are measured in the same wire. So, the measured voltage is added with the potential difference created into the wires (Cause of Ohms Law : U=RI). For high resistivity (from 1M ohm/sq), this method can be used because contact and spreading resistance are negligible. For low resistivity measurement, this method will be not accurate because contact and spreading resistance will very close to sample resistance. That why, for low resistivity, the best method is the 4 point probing. Current is sent in two probes and voltage is measured by two other probes. So, the measured voltage is really that circulate into the sample with no current. So there is no potential difference into the wires and contact and spreading resistance are not high. Measure will be more accurate.
FT09 MW / Page 3 of 9 Practical Measurements : It is not practical to cut the material into little squares and contact the end surfaces. A much more practical and useful measurement of Rs is made using a 4 point probe where the 4 probes contact the top surface of the film or material. The 4 points are in a line or linear and have a constant spacing (S). In the classical measurement, Current (I) passes through the two end probes and voltage (V) is measured between the two center probes. Probes 1 2 3 4 Current Flux lines Figure 2 The current flowing and the voltage measured are related to Rs in accordance with Maxwell s field equations. (See Figure 2) In the ideal case with : 1. the conducting sheet infinite in the horizontal directions and 2. the thickness is < 0.4 x the spacing of the probes (S) 3. and if the measurement temperature is 23 C and 4. the probe spacing (S) is truly constant, the current and voltage are related to the average resistance by a special solution to the equations : Rs = п/ln (2) x V/I = 4.532 x V/I (ohms/sq) (5) Where п = 3.14159 and ln (2) is the log to the base of 2 = 0.693147 and resistivity = Rs x t (ohm-cm) (6) Therefore the resistivity may be calculated from the measurements of V and I and film thickness (t). Conversely if the resistivity of the material is known, the thickness may be calculated from t = resistivity / Rs (cm) (7)
FT09 MW / Page 4 of 9 Correction Factors : As a practical consideration the conditions for equation (5) are not usually in force. However, the four point probe technology is useful for process control. An accuracy approaching 0.5% is desired. Therefore equation (5) is modified using correction factors as follows : Rs = Geometry CF x Thickness CF x Temp CF x (V/I) (8) The Geometry CF is best calculated and measured using the Dual Configuration technique and has a nominal value of 4.532 (See appendix 1). The Dual Configuration method also corrects for variance in the probe tip spacing (S). The thickness correction factor (TCF) can become important for practical measurements of substrates. TCF is a calculation of the correction to the V/I reading required to obtain the true Rs as a function of the ratio of (t/s) or film thickness (t) to the spacing of the probes (S). (See Appendix 2) The Temp CF normalizes the readings to 23 o C and may be found by measuring the temperature (T) and looking up the temperature coefficient of resistance (TCR) for the measured material. (See Appendix 3) The Lucas Signatone and FDI Resistivity Measurement equipment uses the appropriate correction factors to automatically report the correct values of Rs and resistivity or film thickness normalized to 23 o C, with the effects of thickness and horizontal geometries included. (temperature factor also include on special equipments) The methods used are discussed in the appendices and may be reviewed in the referenced source material.
FT09 MW / Page 5 of 9 Appendix 1. Geometry Correction Factor & the Dual Configuration Method Introduction : Consider Figure 2 showing the current flux lines flowing from Probe 1 to Probe 4 with voltage measured between Probe 2 and Probe 3. Now consider one has a non conducting boundary and non constant probe spacing. Note the current flux lines will be constrained by the introduction of the non conductive boundary and the solution to the field equations as given in equation (5) is not accurate. Also, partly because of the error in spacing, the voltage measured between probes 2 and 3 will be in error as compared to the ideal of equation (5). Dual Configuration Testing : A Dual Configuration method for correcting the readings is given in NIST (National Institute of Standards and Technology) Special Publication 260-131 pages 11-17. This method utilizes two sets of readings with different probe connections to the current source and volt meter, then using the ratio of the results as shown in equation (11) below. The errors due to Geometry considerations are greatly reduced by this method giving accuracy typically in the order of 0.5%. The method is valid for geometries with non conducting boundaries more than 3 x the probe spacing (S). Because probe spacings are generally small, equation (11) is very useful for most practical applications. I I 1 2 V 3 4 V 1 2 3 4 First configuration Second configuration When using the Dual Configuration technique the constant 4.532 or П/ln (2) of equation (5) is replaced by the Geometry CF or Ka in equation (8). Ka is calculated from the results of two measurements as follows : Measurement 1 : Current is passed between pins 1 and 4 and voltage is measured between pins 2 and 3. Then Ra is calculated by Ra = V 23/I 14 (ohms) (9) Measurement 2 : Current is passed between pins 1 and 3 and the voltage is measured between pins 2 and 4. Rb is calculated by Rb = V 24/I 13 (ohms) (10) Then the Geometry CF, Ka, is calculated by Ka = -14.696 +25.173 x (Ra/Rb) -7.872 x (Ra/Rb) 2 (11)
FT09 MW / Page 6 of 9 Appendix 2. The Thickness Correction Factor The solution to the ideal equation (5) assumes the current is constrained to stay near the surface in a thin film. Therefore the electrons are forced to follow the field lines predicted by Maxwell s equations. However as the conducting layer becomes thicker relative to the probe spacing the assumption is not valid. The true value of Rs must be adjusted a thickness correction factor TCF as noted below. (See ASTM Standard Designation : F 84-99, pages 10 13) The thickness correction factor TCF may be calculated as a function of the conductive layer thickness / probe tip spacing (t/s) as follows : For t/s < = 0.4 For 0.4 < t/s < = 3.45 Where : TCF = 1 (12) TCF = 1/D(t/S) x 1.3863 / (t/s) (13) M D(t/S) = 1 + 2 {[1/4 + (n x t/s) 2 ] -1/2 [1 + (n x t/s) 2 ] -1/2 } n=1 N + [3/4 x (1/n(t/S)) 3 45/64 x (1/n(t/S)) 5 + 315/512 x (1/n(t/S)) 7 ] n=m+1 Where M = integer (2 x 1/ (t/s) +1) N is the smallest value of n for which the increment in the second summation is less than 10-5 The values for TCF for this range are shown below : Ratio t/s Thick CF 0.5 0.998161 0.6 0.992574 0.7 0.982332 0.8 0.966831 0.9 0.94645 1 0.921956 1.1 0.894378 1.2 0.864725 1.3 0.833992 1.4 0.802958 1.5 0.772222 1.6 0.742216 1.7 0.7132 1.8 0.685429 1.9 0.65897 2 0.633832 2.1 0.610077 2.2 0.587616 2.3 0.566431 2.4 0.546482
FT09 MW / Page 7 of 9 2.5 0.527658 2.6 0.509916 2.7 0.493203 2.8 0.477415 2.9 0.46251 3 0.448425 3.1 0.43512 3.2 0.422509 3.3 0.41056 3.4 0.399227 3.5 0.388468 3.6 0.378256 3.7 0.36853 3.8 0.359269 3.9 0.350443 4 0.342024 For t/s > 3.45 TCF = 2 x ln(2) / (t/s) = 1.386 / (t/s) (14) The derivation of (14) is a result of assuming TCF will approach the correct value for the equation for resistivity where t/s is large. This equation is : resistivity = 2 п x S x (V/I) (15) (See VLSI Technology, by S. M. Sze, Published by McGraw-Hill, p 32) Note : The value of the probe spacing (S) is sometimes chosen to be 0.625 inch so that equation 15 reduces to resistivity = 2 п S x (V/I) = 2 x 3.1415 x.0625 x 2.54 x (V/I) = (1) x V//I (16) Also note that the value calculated is for resistivity not Rs. Then from equation (4) Rs = resistivity / t = 2 п x S/t x (V/I) (17) Combining equations (17) and (5) and assuming large Geometry dimensions and temperature at 23 o C Rs = 2 п x S/t x (V/I) = п/ln(2) x TCF x (V/I) (18) Solving for TCF gives equation (14) TCF = 2 x ln(2) / (t/s) = 1.386 / (t/s) (14)
FT09 MW / Page 8 of 9 Appendix 3. Temperature Correction Factors When used for process control the measurement of Rs is normalized to the value it would be if measured at 23 o C as follows : (See ASTM Standard Designation : F 84-99, pages 10 11) Temp CF = (1 TCR x (T-23)) (20) Where T is the measured temperature in o C and TCR is the Temperature Coefficient of Resistance and is a known physical property for most materials. The TCR varies between.01% per o C to.6% per o C for semiconductor material and much less for most other films. Note that the Temp CF equals 1 at 23 o C. The TCR varies dramatically with impurity concentration for semiconductors and less so for metals. Lucas Signatone offers Four-Point Probe based equipment for measuring TCR for temperatures up to 500 o C. The TCR for various impurity types and concentrations silicon is given in Table 5 of ASTM Standard F 84-99, page 11 which may be purchased through SEMI. Contact us for help. Appendix 4. References The Certification of 100 mm Diameter Silicon Resistivity SRMs 2541 through 2547 Using Dual- Configuration Four-Point Probe Measurements, J. R. Ehrstein and M. C. Croarkin. *NIST SPECIAL PUBLICATION 260-131 Standard Test Method for Measuring Resistivity on Silicon Wafers with an In-Line Four-Point Probe. *ASTM Standard F 4-99 Standard Test Methods for Conductivity Type of Extrinsic Semiconducting Materials. *ASTM Standard F 42-93 Standard Test Method for Sheet Resistance of Thin Metallic Films with a Collinear 4 Probe-Array. *ASTM Standard F 390 78 (Re-approved 1991) Standard Test Method for Sheet Resistance of Silicon Epitaxial, Diffused, Polysilicon, and Ion- Implanted Layers Using an In-Line Four-Point Probe. *ASTM Standard F 374 94a
FT09 MW / Page 9 of 9 Appendix 5. How to choose the appropriate probe head? One probe head is characterized buy 5 things : Spacing between tips Radius of the tips Pressure applicated on the tips Material of the tips Termination with connector 1 Spacing The spacing is chosen according the sample size. We need to be in accordance with the law of resistivity (for single method, sample must be 10xS and for dual method sample must be 3xS minimum). For the best accuracy we should have the highest spacing possible. 2 Radius and Pressure The radius and the pressure are linked. All depend of the sample, thickness, material... Pressure is made by springs calibrated to always applicate the same pressure, that probe is just on contact, or it penetrate the layer. Each tips have independent spring, to be able to applicate the same pressure even if the surface is not totally plane or if the probe head is not totally align. To choose the best probe head, we use the repeatability method. The principle consist to do four or five measurements very close one of the other one on the sample, and found the best repeatable results with several probe heads (ration radius/pressure). This ration is very important to be accurate. If this action is not correct, four case could be happen : Radius or pressure are too hard, tips will penetrate totally the layer and we will characterized the boundary substrate. Radius or pressure are too hard, tips will penetrate the layer but without through it. Properties will be modified, and measurement will be not accurate. Radius or pressure are too soft and an oxide layer protect the sample, tips couldn't penetrate the oxide layer, so we will characterized the oxide layer. Radius or pressure are too soft, tips will have not good contact with the material, so measurement will be not accurate. For the best accuracy, we should have the finest radius, and the highest pressure possible. 3 Material Osmium is harder than tungsten. It is nearly as hard as a diamond. Therefore, it will last longer. As far as best uses, really Osmium and Tungsten Carbide are quite interchangeable with only a couple of exceptions. We know that Osmium is best for contacting GaAs wafers. Tungsten Carbide has a difficult time getting good contact but Osmium works well. Conversely, we seem to get better,more consistent results on metals when using Tungsten Carbide, but Osmium does still work.