# Recitation 2: Equilibrium Electron and Hole Concentration from Doping

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1 Recitation : Equilibrium Electron and Hole Concentration from Doping Here is a list of new things we learned yesterday: 1. Electrons and Holes. Generation and Recombination 3. Thermal Equilibrium 4. Law of Mass Action 5. Doping - (donors and acceptors) and charge neutrality 6. Intrinsic Semiconductor vs. Extrinsic Semiconductor 7. Majority and Minority carriers 1 Electrons and Holes This refers to the free electrons and holes. They carry charges (electron -ve and hole +ve), and are responsible for electrical current in the semiconductor. Concentration of electron (= n) and hole (= p) is measured in the unit of /cm +3 or cm 3 (per cubic centimeter). Remember in Si the atomic density is 5 10 cm 3, very useful number Generation and Recombination Generation is one way to obtain free e & h in semiconductors. 1 electron-hole pair (1e + 1h) is generated by breaking a bond. Recombination is the reverse process. 3 Thermal Equilibrium A concept which will be used very often. Thermal equilibrium is defined as steady state + no extra energy source. Note that we have generation or recombination under thermal equilibrium. It is just that the two rates are equal and cancel each other, so the concentrations of e & h do not change. n o and p o refer to concentrations in thermal equilibrium. 4 Law of Mass Action At each temperature T, under thermal equilibrium: n o p o = constant = f(t ) (only depends on temperature) n o p o = n i (T ) (n i intrinsic carrier concentration) 1

2 This is like a chemical reaction: H 0 H + +OH [H + ][OH ]=10 14 (mol/l) at Room T bond e +h + n o p o = n i (T )=10 0 (cm 3 ) at Room T Note n i has a temperature dependence: E G n (T ) 3/ i = A e k B T A is a constant, T is in Kelvin, T (K) = 73 + T( C), and k B = ev/k. E G is the Bandgap energy of the semiconductor - it also corresponds to the ease of bond breakage. For Si, E G =1.1 ev. Example 1 At room temperature, T = 300 K, n i (300 K) = cm 3. What is n i (500 C)? n i (500 C) = n i (773 K) n i (773 K) n i (300 K) = E G ( ) 773 3/ e kb (773) e 8.4 =4.14 = E G e 1.65 e kb (300) Therefore, n i (773 K) = n i (300 K) = cm 3 = cm 3 Something to observe: At room temperature, n o = p o =10 10 cm 3 for Si. Atomic density is 5 10 cm 3. There fore, only a tiny fraction of atoms ( 5 10 = = %) lose an electron in one of their 4 bonds. By heating up to 500 C, the concentration of free carriers goes up 10 6 (1 million) times, but the percentage is still quite low.

3 5 Doping and Charge Neutrality Doping free electrons generated by As n type dopants: As, P, Sb give out an electron easily leave behind a positively charged ion Donor concentration N d (cm -3 ) a hole generated due to bond breaking and e - given to B - p type dopants: B have one less electron, will grab one easily from another place, become negatively charged and thus generate a hole Acceptor concentration N a (cm -3 ) Figure 1: Types of Doping Charge neutrality Although foreign atoms are introduced in Si, the overall semiconductor is charge neutral. Therefore, concentration of positive charges = concentration of negative charges. The positive charges include holes (p) and donors (N d ). The negative charges include electrons (n) and acceptors (N a ). p o + N d n o N a =0 3

4 Example Boron doping, dopant concentration cm 3. At R.T. under thermal equilibrium N d =? N a =? n o =? p o =? n i =? p or n type? Boron is an acceptor meaning N a =10 17 cm 3, N d =0. n i = cm 3 at R.T. under thermal equilibrium (material property, doping does not matter) p o N a =10 17 cm 3 (because N a n i ) n o p o = 10 0 cm cm 6, n o = cm 3 =103 cm 3 6 Intrinsic Semiconductor vs. Extrinsic Semiconductor In the above example, the semiconductor is extrinsic because the carrier concentrations are determined by the dopant concentrations. Example 3 Si at 500 C, with As doping cm 3, extrinsic or intrinsic? At 500 C, n i (773 K) = cm 3 > N d It is intrinsic semiconductor even though there is doping. Example 4 A semiconductor can have both dopings. If N a =10 15 cm 3, N d =10 19 cm 3 = n-type Si, n o p o even though we have N a =10 15 cm 3. When things get complicated, the following relations always work: p o + N d n o N a = 0 n o p o = n i (T ) 4

5 Consider example, N d =0N a =10 17 cm 3. We said p o N a =10 17 cm 3. How accurate is this approximation? n o p o = n i (T ) n i (T ) = n o = plug into p o n i (T ) + Nd N a p o = 0 p o N a p o n i = 0 = N a N a 1+ 4n i (T ) p o ± Na discard, otherwise p o < 0 = p o = N a N a n i (T ) Na p o charge neutrality i (T ) a p is a good approximation since 4n o N a 1 N 7 Majority and Minority Carriers In example, p o N a =10 17 cm 3 n o =10 3 cm 3. Hole is majority carrier and electron is minority carrier. 5

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