BASICS OF SEMICONDUCTOR LECTURE 6 BASICS OF SEMICONDUCTOR PHYSICS Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

DENSITY OF STATES Current is due to the flow of charge carriers To know the number of electrons and holes in semiconductor is important in order to calculate the current flow Number of electrons and holes as a function of available energy levels Based on Pauli’s exclusion principle, 1 electron per energy level Electrons and holes concentration are calculated based on the calculation of density of allowable energy levels or density of states DENSITY OF STATES IS CALCULATED BY SOLVING FREE ELECTRON IN 3-DIMENSIONAL INFINITE POTENTIAL WELL USING SCHRODINGER WAVE EQUATION. Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

DENSITY OF STATES PER UNIT VOLUME OF THE CRYSTAL ________ √ E ……… (1) g (E) = h3 Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. As the energy of this free electron becomes small, the number of available states decreases. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

Example Consider the density of states for a free electron given by Equation 1. Calculate the number of states per unit volume with energies between 0 and 1 eV. 1 eV 1 eV 4 π (2m)3/2 ∫ ∫ ________ N = g (E) dE = √ E dE h3 4 π (2m)3/2 ________ . 2/3 . E3/2 N = h3 Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. 4 π [ 2 (9.11 x 10-31)]3/2 __________________ N = . 2/3 . (1.6 X 10-19) 3/2 (6.625 x10-34)3 = 4.5 x 1027 states / m3 = 4.5 x 1021 states / cm3 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

STATISTICAL MECHANICS THREE DISTRIBUTION LAWS DETERMINING THE DISTRIBUTION OF PARTICLES AMONG AVAILABLE ENERGY STATES MAXWELL-BOLTZMANN PROBABILITY FUNCTION PARTICLES ARE CONSIDERED AS TO BE DISTINGUISHABLE GAS MOLECULES IN A CONTAINER AT LOW PRESSURE BOSE-EINSTEIN FUNCTION PARTICLES ARE INDISTISTINGUISHABLE BLACK BODY RADIATION FERMI-DIRAC FUNCTION PARTICLES ARE INDISTINGUISHABLE ELECTRONS IN CRYSTAL Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

FERMI-DIRAC PROBABILITY FUNCTION GIVES THE PROBABILITY THAT A QUANTUM STATE AT ENERGY E WILL BE OCCUPIED BY AN ELECTRON 1 _________________ F (E) = ….. (2) 1 + exp (E – EF) / kT k – Boltzman constant = 1.38 x 10-23 J/K, T in K Where EF is Fermi Energy, and defined as the energy where the probability of a state being occupied is 0.5 or 50%. Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. 1 1 1 _________________ _________ _________ F (EF) = = = 1 + exp (EF – EF) / kT 1 + exp (0) 1 + 1 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

DISTRIBUTION FUNCTION VS ENERGY LET T = 0K AND E < EF 1 1 _________________ _________________ = = 1 1 + exp (E – EF) / kT 1 + exp (-∞) F (E < EF) = 1 LET T = 0K AND E > EF Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. 1 1 _________________ _________________ = = 0 1 + exp (E – EF) / kT 1 + exp (+∞) F (E > EF) = 0 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

a quantum states being occupied in energy levels E1 through E4 is Consider a system of 13 electrons with discrete energy levels as below; At T = 0 K E5 E4 E3 E2 E1 Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. The electrons will be in the lowest energy states, so the probability of a quantum states being occupied in energy levels E1 through E4 is unity. The probability of quantum states to be occupied in energy level E5 is zero In this case, Fermi energy must be above E4 but less than E5 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

Consider a system of 13 electrons with discrete energy levels as below; At T > 0 K, E = EF E5 E4 E3 E2 E1 Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Electrons gain certain amount of thermal energy so that some can jump to higher energy levels. As the temperature change, the distribution of electrons vs energy changes 1 1 1 _________________ _________ _________ F (EF) = = = = 0.5 1 + exp (EF – EF) / kT 1 + exp (0) 1 + 1 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

Let T = 300K. Determine the probability that an energy level 3kT above Example Let T = 300K. Determine the probability that an energy level 3kT above the Fermi energy is occupied by an electron. 1 _________________ F (E) = 1 + exp (E – EF) / kT 1 _________________ = 1 + exp (3kT / kT) 1 Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. _________ = 1 + 20.09 = 4.74 % Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

CONSIDERING A GENERAL CRYSTAL AND APPLYING SUMMARY CONSIDERING A GENERAL CRYSTAL AND APPLYING THE CONCEPT OF QUANTUM MECHANICS TO DETERMINE THE CHARACTERISTICS OF ELECTRON IN SINGLE-CRYSTAL LATTICE. TO APPLY THOSE CONCEPT TO A SEMICONDUCTOR MATERIALS IN THERMAL EQUILIBRIUM DENSITY OF STATES IN CONDUCTION AND VALENCE BANDS TO BE APPLIED WITH FERMI-DIRAC FUNCTION TO DETERMINE THE CONCENTRATIONS OF ELECTRONS AND HOLES IN THE RESPECTIVE BANDS. Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

ELECTRONS AND HOLES DISTRIBUTION IN EQUILIBRIUM THE ELECTRONS DISTRIBUTION IN CONDUCTION BAND IS GIVEN BY; n (E) = gc (E) F (E) ….. (3) Where gc (E) is density of states in the conduction band F (E) Fermi-Dirac distribution function Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Total electron concentration per unit volume in the conduction band can be calculated by integrating Equation (3) over entire conduction band energy. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

THE HOLE DISTRIBUTION IN VALENCE BAND IS GIVEN BY; p (E) = gv (E) [ 1- F (E)] ….. (4) Where gv (E) is density of states in the valence band [1-F (E)] the probability of states not occupied by electron Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Total hole concentration per unit volume in the valence band can be calculated by integrating Equation (4) over entire valence band energy. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

INTRINSIC SEMICONDUCTOR pure s/c material with no impurity atoms and no lattice defect At T = 0 K, all states in valence band are filled with electrons, states in conduction band are empty. Fermi energy is between Ec and Ev As the temperature increase above 0 K, few electrons will get enough energy to jump to conduction band and creating holes in the valence band. In intrinsic material, electrons and holes are created in pairs by thermal energy. So the number of electrons in conduction band is equal to the number of holes in the valence band Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

gc gv Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

[ ] ∫ Electron concentration at thermal equilibrium given by; ∞ ni = gc (E) F (E) dE Ec [ -(Ec – EFi ) ] _________ ……….. (5) ni = Nc exp kT Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Where Nc effective density of states in the conduction band Nc at T = 300K Si GaAs Ge 2.8 x 1019 cm-3 4.7 x 1017 cm-3 1.04 x 1019 cm-3 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

[ ] ∫ Holes concentration at thermal equilibrium given by; pi = Ev ∫ pi = gv (E) [1 - F (E)] dE - ∞ -(EFI – Ev ) [ ] _________ ……….. (6) pi = Nv exp kT Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Where Nv effective density of states in the valence band Nv at T = 300K Si GaAs Ge 1.04 x 1019 cm-3 7.0 x 1018 cm-3 6.0 x 1018 cm-3 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

In intrinsic semiconductor, electron concentration is equal to the hole concentration, thus ni = pi and nipi = ni2 (MASS ACTION LAW) – the product of n p is always a constant for a given semiconductor material at given temperature If we take the product of Equation 5 and 6, [ -(Ec – EFi ) -(EFI – Ev ) ] [ ] _________ _________ ni2 = Nc Nv exp . exp kT kT Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. [ ] -(Ec – Ev ) = Nc Nv exp _________ kT -Eg = ______ Nc Nv exp ….. (7) kT Nc and Nv vary at T3/2 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

band. The value of Nc for silicon at T=300K is Nc = 2.8 x 1019 cm-3 Example Calculate the probability that a state in the conduction band is occupied by an electron and calculate the thermal equilibrium electron concentration in silicon at T=300K. Assume Fermi Energy is 0.25eV below the conduction band. The value of Nc for silicon at T=300K is Nc = 2.8 x 1019 cm-3 The probability that an energy state at E = Ec is occupied by electron is given by Fermi-Dirac probability function 1 [- (Ec – EF) ] _________________ _________ F (Ec) = exp Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. 1 + exp (Ec – EF) / kT kT For electrons in the conduction band, E > Ec. If (Ec – EF) >> kT, then (E – EF) >> kT. So the Fermi function can be reduced to Boltzman Approximation Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

[- (Ec – EF) ] _______ F (Ec) = exp kT kT = 1.38 x 10-23 J/K . 300K = 4.14 x 10-21 J kT = 4.14 x 10-21 / 1.6 x 10-19 eV = 0.0259 eV Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. [- 0.25 ] _______ F (Ec) = exp = 6.43 x 10-5 0.0259 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

[ ] The electron concentration is given by -(Ec – EFi ) _________ Nc exp kT = (2.8 x 1019) exp (-0.25 / 0.0259) = 1.8 x 1015 cm-3 Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1

at T = 400K. Assume Fermi level is 0.27 eV above the valence band. The Example Calculate the thermal equilibrium hole concentration in intrinsic silicon at T = 400K. Assume Fermi level is 0.27 eV above the valence band. The value of Nv for silicon at T=300K is Nv = 1.04 x 1019cm-3 At T = 400K kT = 0.0259 (400/300) = 0.03453 eV Nv = (1.04 x 1019) (400/300)3/2 = 1.60 x 1019 cm-3 Semiconductor Manufacturing Technology It has been fifty years since the invention of the transistor. The technology behind this topic cannot be addressed adequately in a single presentation, but has been separated into four modules: Module 1: Basic Principles Module 2: Transistor Design and Manufacturing Overview Module 3: Semiconductor Manufacturing Processes Module 4: Semiconductor Economics In this third module, we will review the key semiconductor processes, materials and equipment used to fabricate devices, and discuss process conditions and chemistry. pi = (1.6 x 1019) exp ( -0.27/0.03453) = 6.43 x 1015 cm-3 Praxair Semiconductor Manufacturing Technology, Module 3: Semiconductor Manufacturing Processes 1