LAB III. CONDUCTIVITY AND THE HALL EFFECT 1. OBJECTIVE The conductivity, σ, of a silicon sample at room temperature will be determined using the van der Pauw method. The Hall Effect voltage, VH, and Hall coefficient, RH, for the same sample will be measured using a magnetic field. These measurements will enable the student to determine: the type (n or p) and doping density of the sample as well as the majority carrier’s “Hall mobility.” 2. OVERVIEW You will use the van der Pauw method to determine the conductivity of your silicon sample in the first section of this lab. You will find the Hall voltage and coefficient in the second section. These measurements will be used to find the semiconductor type (n or p), the doping density, and the majority carrier mobility (Hall mobility) of the silicon sample. Information essential to your understanding of this lab: 1. An understanding of material types (e.g,, n-type, p-type). 2. The relationship between current density and (carrier density/mobility/conductivity). Materials necessary for this Experiment 1. Standard testing station 2. One mounted Silicon chip 3. Two magnets ~ 0.0125 Weber / m2 (1 Weber = 1 V-s) (Use them to sandwich the sample.) 3. BACKROUND INFORMATION 3.1. CHART OF SYMBOLS Table 1. Symbols used in this lab. Symbol Symbol Name Units h+ Hole Positive charge particle e- electron Negative charge particle q magnitude of electronic charge 1.6 x 10-19 C p hole concentration (number h+ / cm3) n electron concentration (number e- / cm3) ni intrinsic carrier concentration 3 ND Donor concentration (number donors / cm ) 3 NA Acceptor concentration (number acceptors / cm ) -23 kb Boltzmann's constant 1.38 x 10 joules / K T temperature K Eg Energy gap of semiconductor eV J Current density A / cm2 E Electric field V / cm conductivity ( - cm)-1 resistivity - cm mobilities cm2 / V-sec vd drift velocity cm / sec B magnetic field Weber / m2 -3 Nv valence band effective density of states cm -3 Nc conduction band effective density of states cm 3-1 3.2. CHART OF EQUATIONS Table 2. Equations used in this lab. Equation Name Formula 1 Intrinsic carrier equation 2 (Eg / 2kT ) ni Nc Nve 2 Charge Neutrality p + ND = n + NA 2 3 Law of Mass Action pn = ni (T) 4 Current density J J n J p nE p E 5 Conductivity due to n n qn n 6 Conductivity due to p p q p p 7 Conductivity of a material 1 q n q p n p 8 Resistivity formula for the t R12,34 R23,41 van der Pauw F ln 2 2 9 Resistivity formula in terms 1 of sheet resistance R t s 10 Current density J x p Ex 11 Drift velocity vx p Ex 12 Lorentz force (y-direction) FB = qvx x Bz 13 Induced Hall Field ( Ey ) qE y qvx Bz 14 E-field equation for the 1 semiconductor sample E J B y qp x z 15 Hall coefficient 1 R H qp 16 Induced E-feild Ey RH J x Bz 17 Hall voltage VH Ey w 18 Current density J x I x /tw 19 Derivation of the carrier 1 I B density in a p-type material p x z q t VH 20 Derivation of Hall coefficient VH t RH I x Bz 21 Derivation of the mobility p R p qp H p 3-2 3.3. CONDUCTIVITY OF A SEMICONDUCTOR One of the most basic questions asked in semiconductor devices is “what current will flow for a given applied voltage”, or equivalently “what is the current density for a given electric field” for a uniform bar of semiconductor. The answer to this question is a form of Ohm's law, namely: J = Jn + Jp = σn*E + σp*E (4) Here, J is the current density (A/cm2) E is the applied electric field (V/cm) and σ is the conductivity (1/(Ohm-cm)). The n and p subscripts refer to electrons and holes. This equation tells us that the total current density J is equal to the sum of the electron and hole current densities. Those are given by the electron conductivity * E plus the hole conductivity * E. Note that the conductivity increases as the numbers of electrons and holes increases. σn = q*n* μn (5) σp = q*p* μp (6) σ = σn + σp = q*n* μn + q*p* μp (7) Remember q = 1.6 x 10-19 Coulombs and n and p are electron and hole densities (number per cm3). 2 The quantities μn and μp are called the “electron and hole mobilities” respectively (cm /V-sec). Mobilities describe the average velocity (m/s) per unit electric field that electrons or holes experience as they propagate through the lattice of the semiconductor. In fact, we write that the electron and hole average velocities are defined as vn = μn*E and vp = μp*E. Note that the conductivity of a semiconductor depends upon both the carrier densities and their mobilities. Consequently, a simple measurement of the conductivity alone can only be used to find n and p if μn and μp are already known. (Note: The values of n and p are related by the law of mass 2 action (n*p = ni ) and so we only need to know either n or p to know them both.) It would be nice if the mobilities were simple constants, but they are not. μn and μp are functions of temperature as well as the doping concentrations (NA and ND) and therefore functions of n and p! Fortunately, we know these functions from many prior calibration measurements done by scientists worldwide. So a simple measurement of conductivity still can be used to give us an estimate of n and p provided we at least know the semiconductor type (n-type or p-type). We use the Irwin curves to make the connection between the semiconductor conductivity and its’ doping density (NA or ND). Note that 2 2 for p-type material NA ~ p and n ~ ni /NA; for n-type material ND ~ n and p ~ ni /ND. Figure 1 shows the Irwin curves. It is a plot of the silicon conductivity as a function of either NA or ND assuming that the other is equal to zero. You should familiarize yourself with it. There are many measurement methods to find the conductivity of a sample. One of the most common methods is called the “four-point probe method.” We will be doing a variant of the four- point probe method called the van der Pauw method in this lab. It is a measurement method for arbitrarily-shaped samples. You will compare these results with measurements of the Hall Effect. The van der Pauw method is commonly used to measure the conductivity of semiconductors, particularly for thin epitaxial layers grown on semi-insulating substrates. You will do this too. Using this method, you will measure the conductivity of your sample. Once you find the resistivity of the sample, you can use the Irwin curves to estimate the values of n or p. For example if you had a conductivity of 0.1 mho/cm (resistivity is 10 ohm-cm), NA would be approximately 1.3 x 1015 cm-3. Since the van der Pauw method cannot allow you to determine the type of your semiconductor sample, you’ll have to use the Hall Effect to find that first. 3-3 Resistivity N-type Resistivity P-type Resistivity 1.E+04 1.E+03 1.E+02 p-type 1.E+01 n-type 1.E+00 Resistivity (Ohm-cm) 1.E-01 1.E-02 1.E-03 1.E+12 1.E+13 1.E+14 1.E+15 1.E+16 1.E+17 1.E+18 1.E+19 1.E+20 Dopant Concentration (cm-3) Figure 1. Irwin curves for singly doped silicon at 300 K. 3.4. THE VAN DER PAUW METHOD L. J. van der Pauw proved that the resistivity of an arbitrarily shaped sample could be estimated from measurements of its resistance provided the sample satisfied the following conditions: 1) contacts are at the boundary; 2) contacts are small; 3) the sample is uniformly doped and uniformly thick; 4) there are no holes in the sample. He derived a correction factor, f, to use in that estimation. In the van der Pauw method, and in all 4-point probe methods, a current is forced between two contacts (call them contacts A & B) while the voltage is measured between two different contacts (C & D). It is often the case that UG students wonder why the voltage is NOT measured between contacts A & B. The thought is: “Wouldn’t you get the resistance by simply dividing VAB by IAB?” The answer is: You would get a resistance, but it would be the WRONG resistance. The resistance that is correct is VCD/IAB. VAB/IAB gives too large a resistance because it always includes something called the “contact resistance” too. The contact resistance is a resistance that sits exactly at the contact between the metal probe (the contact) and the semiconductor. This resistance has a voltage drop across it whenever there is current flowing through it (Vcontact=IAB*Rcontact.) The problem is 3-4 that this contact resistance has nothing to do with the semiconductor conductivity! Therefore, we do not want to have Vcontact be part of our measured voltage. We want only the voltage caused by the conductivity of our sample to be divided by IAB. Figure 2 shows a schematic diagram of this problem.