Charge Carriers in Semiconductor To understand PN junction s IV characteristics, it is important to understand charge carriers behavior in solids, how to modify carrier densities, and different mechanisms of charge flow. After notes by Wentai Liu- UCSC
Silicon - Semiconductor Conductor low resistivity Insulator high resistivity Semiconductor - intermediate Si has four valence electrons. Therefore, it can form covalent bonds with four of its neighbors. When temperature goes up, electrons in the covalent bond can become free.
Electron-Hole Pair Interaction With free electrons breaking off covalent bonds, holes are generated. Holes can be filled by absorbing other free electrons, so effectively there is a flow of charge carriers. Conduction electrons are negatively charged Holes are positively charged. Concentration of holes and electrons can be modulated, for example, by impurity dopants
Free Electron Density in Intrinsic Silicon E g, or bandgap energy determines the amount of energy to break off an electron from its covalent bond. There exists an exponential relationship between the free-electron density and bandgap energy. n n n i i i = 5.2 10 ( T ( T = = 300 600 0 0 15 T K ) K ) E exp 2kT = 1.08 10 3/ 2 = 1.54 10 g 10 15 electrons electrons electrons / / / cm cm cm 3 3 3
Extrinsic by Doping Pure Si can be doped with other elements to change its electrical properties. If Si is doped with P (phosphorous), each P atom can contribute a conduction electron, then it has more electrons, or becomes N-type If Si is doped with B (boron), each B atom can contribute a hole. Then it has more holes, or becomes P-type Donor Group V Acceptor Group III
Summary of Charge Carriers
How to use energy band diagrams? Remember two features There is more than 1 free electron (hole) in a semiconductor. They cannot have all the same energy, thus the available free electrons (n) or holes (p) will be distributed in energy. Potential barriers (due to bands that are bended) across a junction can block the carriers from going from one side to the other side when the barrier is higher in energy than the highest (in energy) available carrier that wants to diffuse or drift across the barrier. External fields lower or increase these potential barriers letting more or less carriers from the distribution through.
Fermi-Dirac distribution and the Fermi-level Density of states tells us how many states exist at a given energy E. The Fermi function f(e) specifies how many of the existing states at the energy E will be filled with electrons. The function f(e) specifies, under equilibrium conditions, the probability that an available state at an energy E will be occupied by an electron. It is a probability distribution function. EF = Fermi energy or Fermi level k = Boltzmann constant = 1.38 10 23 J/K = 8.6 10 5 ev/k T = absolute temperature in K 1
Fermi-Dirac distribution: Consider T 0 K For E > E F : f 1 ( E EF) 1 exp ( ) 0 For E < E F : f 1 ( E EF) 1 exp ( ) 1 E E F 0 1 f(e) 2
Fermi-Dirac distribution: Consider T > 0 K If E = E F then f(e F ) = ½ If E EF 3kT E E then exp F 1 kt ( E E Thus the following approximation is valid: f E) exp kt i.e., most states at energies 3kT above E F are empty. ( F ) If E EF E E 3kT then exp F 1 kt Thus the following approximation is valid: So, 1 f(e) = Probability that a state is empty, decays to zero. So, most states will be filled. f ( E) 1 exp kt (at 300 K) = 0.025eV, E g (Si) = 1.1eV, so 3kT is very small in comparison. E E kt F 3
Temperature dependence of Fermi-Dirac distribution 4
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Equilibrium distribution of carriers Distribution of carriers = DOS probability of occupancy = g(e) f(e) (where DOS = Density of states) Total number of electrons in CB (conduction band) = n 0 E E top C gc( E) f ( E) de Total number of holes in VB (valence band) = p 0 E V g ( E) 1 E Bottom V f ( E) de 6
Electron and Hole Density The product of electron and hole densities is ALWAYS equal to the square of intrinsic electron density regardless of doping levels. 2 np = n i Majority Carriers : Minority Carriers : Majority Carriers : Minority Carriers : p N n n N n N n p N A 2 i A D 2 i D
Charge Transport Mechanism - Drift The process in which charge particles move because of an electric field is called drift. Charge particles will move at a velocity that is proportional to the electric field. = = E v E v n e p h µ µ h W J q n h W v dt q n h W x d dt dq I n n n. ))... ( ( = = = = E p n q q p E q n E J p n p n tot ) ( µ µ µ µ + = + = Total drift current
Velocity Saturation v µ 0 µ = 1+ be µ 0 vsat = b µ 0 = µ 0E 1+ v sat E A topic treated in more advanced courses is velocity saturation. In reality, velocity does not increase linearly with electric field. It will eventually saturate to a critical value.
Charge Transport Mechanism - Diffusion Charge particles move from a region of high concentration to a region of low concentration. It is analogous to an every day example of an ink droplet in water. Diffusion current is proportional to the gradient of charge (dn/dx) along the direction of current flow. Its total current density consists of both electrons and holes. I J = n = AqD qd n n dn dx dn dx J J p tot = qd p = q( D n dp dx dn dx D p dp ) dx
Einstein s Relation D = µ kt q While the underlying physics behind drift and diffusion currents are totally different, Einstein s relation provides a mysterious link between the two.