Colloqium problems to chapter 13 1. What is meant by an intrinsic semiconductor? n=p All the electrons are originating from thermal excitation from the valence band for an intrinsic semiconductor. Then the concentration of electrons will equal the concentration of holes. This situation is nearly realized in a very pure semiconductor with no donors or acceptors. 2. What is meant by a degenerate semiconductor? n >= Nc for n-type or p >= Nv for p-type. So the carrier concentration is equal or larger than the effective density of states of the conduction or valence band. That is the same as saying that the concentration of carriers are so large that the average distance between them is comparable to the deBroglie wavelength of the carriers. (In statistical physics terms: The carrier concentration is larger than the quantum concentration. The effective density of states has the same value as the quantum concentration. It is the quantum concentration of the respective band ) 3. Consider a semiconductor. The concentration of electrons, n, in the conduction band is given by " n= # g (E) f (E, E c F )dE Ec Where f(E,EF) is the Fermi-Dirac distribution function. ! What is the units ( or dimension ) of gc(E) ? The dimension of n is #(number) pr volume, for example m-3. Thus the dimension of gc is eV-1m-3 4. Assume we have a Si crystal doped with 1017 P atoms/cm3. The ionization energy of P is taken to be 0.05 eV. a)Calculate at which temperature half of the donor atoms are ionized. b) Calculate the Fermi level at 300 K and 300 °C and 310 °C All these questions could be calculated very much the same way – by getting the relationship between EF, the Fermi level and T the temperature - The general case does not give a simple analytical relationship though, so we will have to calculate numerically. However just for the exercise we choose slight different solution strategies. In all cases we require that we have charge neutrality, set up the equation for that and use the solution of this to calculate what is asked for. Charge neutrality : eq.1 ! ! ! ! n + (NA-= 0) – ND+ - p = 0. Where n, p have their usual meaning and ND+ is the concentration of ionized donors and NA- is that of acceptors. The latter is zero in an extrinsic n-type semiconductor. a) When half of donors are ionized we know the Fermi level goes through the donor level and the Fermi-Dirc distribution has the value 0.5 at ED the donor level. We may also rationalize (alternatively we know from what is written in the book and lectured ..) that this situation will occur at sufficiently low temperature that we can ignore the holes. Equation 1 then becomes eq1 a n = ND/2 We put in an expression for n and have # E " EF & ND E " ED N c exp%" c we put in EF=ED and get T = c (= # 2N & $ kT ' 2 k ln% c ( $ ND ' here we have considered Nc as constant, since it is varying weakly with temperature, We could of course have incorporated the T-dependence of Nc as well b) We here just solve the problem in a general!way. To incorporate the temperature dependence of Nc we use the following expression which will be put into equation eq1: 3 # E c " E F & # 2) me kT & 2 # E " EF & n = N c exp%" eq 2 ( = 2% ( exp%" c ( 2 $ ' $ kT ' $ h kT ' (here me is the electron effective mass) And for the concentration of charged donors we write # & % ( 1 + ( N D = N D (1" f (E)) = N D %1" eq3 % 1+ exp#% E " E F &( ( % ( $ kT ' ' $ Finally for the hole concentration we write. 3 # E F " E v & # 2) mh kT & 2 # E " Ev & p = Nv exp%" ( = 2% ( exp%" F ( where E v = E c " E g eq4 2 $ ' $ kT ' $ h kT ' We put eq2, eq3 and eq4 into eq1 . We will then have an equation with two unknown : EF and T. It is the relationship between them we are interested in, so we pick either EF and calculate T as in problem a) or set/pick T and calculate ! EF as in problem b) Thus for b) equation 1 (eq 1) is written as G(EF) = 0 where G is the functional dependence given by eq1. We can not get an explicit solution, but have do it numerically or iterate towards a solution. (We can use Newtons method for this.) The physical parameters we may use are given in the text or in the book or in lecture notes as m*e=1.08* 9.109 e-31 kg, m*h= 0.56*9.109e-31 kg, h = 6.6e-34*6.2415e18 eVs k=1.38e-23 *6.2415e18 eV/K. ED =0.05 eV, Eg=1.12 eV , ND=1e17*1e6/m3, The answers are a) T=148.646 K, b) EF(300 K)=-0.1282 eV, EF(300 C)=- 0.29277 eV, EF( (310 C)=-0. 29919 eV, The energies are with respect to Ec., the bottom of conduction band A figure of all calculated values is shown below MENA 4000 Coll 13.4 0.1 `E c 0 `E -0.1 -0.2 F Fermi Level,E ( EV) D -0.3 -0.4 -0.5 -0.6 -0.7 0 200 400 600 800 1000 1200 1400 Temperature(K) 5 Assume we have a crystal as in problem 4. We place one end of the crystal in thermal contact with an insulator at 300 C. The other is in contact with an insulator at 310 C. Assume the insulators are very big and have good thermal conductivity – thus their temperatures are 300 and 310 in a stationary state. No net electron flow occurs in the system at a stationary state, but a voltage (potential) difference occur across the Si crystal. This is the Seebeck effect. Make an estimate of the magnitude of the voltage for n-type Si doped to 1017 cm3 at around 305 °C. (Hint: See problem 4) We consider that we have no electron current flowing. The Fermi level is then per definition equal in all parts of the crystal. We have equilibrium with respect to exchange of number of particles/electrons, but we do not have equilibrium with respect to exchange of energy – we have a net energy flowing. We assume however that the state of the different part of the system is approximately equal to that of an equilibrium state for Si at the respective temperatures for different parts. Thus the difference between the conduction band minimum is a function of temperature, but the Fermi level is constant. Then the conduction band minimum varies throughout the crystal. Thus the potential for electrons varies. That means we have a potential difference. This is the sought voltage. See also the following figure. From problem 4 we have EF(300 C)=-0.29277 eV, EF( (310 C)=-0.29919 eV, Thus the potential drop pr degree C is 0.00064 eV = 0.64 mV/° C The Seebeck coefficient is about -0.64 mV/°C 6 When we talk about electrical current in a semiconductor, explain what is the term drift velocity and thermal velocity. Drift velocity is the mean velocity in the direction of the electrical field. There is some confusion about averages of velocity and speed as to which one has a direction and which does not. It is important to separate, but knowing that many people use it somewhat arbitrary, we rather in each case emphasize what we talk about. Without an external electrical field the average velocity considering direction is zero. The average of the magnitude av the velocity, i.e. mean of sqrt(v2) is not zero and without external field the energy comes from random thermal excitations. Calculate a value for the thermal velocity for electrons in Si at room temperature (300 K) Considered as an ideal gas the thermal energy is 3/2 kT this is equal to kinetic energies 1/2mv2 The value of the velocity is thus 3kT 0.026eV *1.6x10"19 kgm 2 v th = = . = 6.5 x104 m/s m*e 1.08x9.109x10"31 kgeVs2 7 The drift velocity of Si saturates at around 106 cm/s. If the conditions where such that we ! had saturation, how long time would it take an electron to move between two regions of a device ( for example from the source to the drain of a field effect transistor) that is 50 nm apart? What is the maximum frequency such device can operate at, only considering the delay of motion? vd = !x/!t => !t = !x/ vd =0.5x10-11 s f=1/t =2x1011 Hz = 200 Ghz; 8 The electron concentration of semiconducting extrinsic Si can be written as # E " EF & n = N c exp%" c ( $ kT ' where Nc is called the effective density of states of the valence band. What is meant by the word extrinsic? n>>p ( in this or p>>n for p-type ) What is meant by the effective density of states? What kind of physical picture lies ! behind it? The word effective is used often for parameters in physics. It is used in the following way. We make an (often, simplistic, idealized) model of the system at hand. This model uses a certain variable. The effective value of this parameter is the number we have to put in order for the parameters predicted by the model to agree with observations. For he effective density of states we make a simple description of the semi-conductor by saying that we have a certain density of states, Nc, located at an energy Ec. Nc is the effective density of states. Notice that in this case we did start out by a more realistic model, but got a mathematical expression for the carrier concentration which is equal to the one from this simplified description. 9 In an electronic semiconductor device the current is transported by diffusion current and by drift current. The electrical field is the driving force for the drift current. What is the driving force for the diffusion current? The driving force for the diffusion current is the concentration gradient 10 Assume a semiconducting Si wafer that is weakly doped with B atoms (1015 at/cm3). We diffuse P atoms into the wafer at a high temperature by subjecting it to a flow of P containing gas molecules. The P concentration at the surface is limited by the solid solubility – say here 1019 at/ cm3. The diffusion time is so long that the p-n junction will occur at a depth of 5 µm. Sketch how the band diagram of this Si crystal varies with depth from the surface and down to 30 µm. Make the sketch semi-quantitatively. ( Hint –estimate calculate the Fermi level at the surface, at the p-n junction and in bulk n-type material.) Discuss the results in relationship to the diffusion of P atoms (during the latter part of the diffusion process) Below is shown the doping concentration vs depth and underneath there is the band diagram versus depth. Since the distance from the Fermi level to the conduction band reflects how large concentration of electrons we have. After the diffusion process the wafer can be brought to equilibrium, then the Fermi level is constant. It then follows that the conduction band minimum varies with postion in the sample. 11 Make a sketch of the density of states vs. energy for an idealized metal. (By idealized metal we consider a metal that is modeled as a gas of non-interacting electrons. ) Draw the position of the Fermi level on the figure. In the above graph the density of states is shown as a black curve. We call it here g(E). It has a E0.5 dependence so g(E) " E 0.5 . The Fermi level is indicated by EF The Fermi level (electrochemical potential) of this idealized metal will decrease as a function of temperature, Argue semi-qualitatively why it decreases ( Hint. The density of ! electrons ( the number of electrons pr unit cell) in the metal does not change with temperature. Sketch how the energy distribution of electrons is and how it changes with temperature. Use known symmetry properties of the Fermi-Dirac distribution function) ! In the sketched figure above are drawn two hypothetical situations showing the electron distributions at two temperatures: 1) At low temperature or T=0, blue curve and 2) at high temperature , red curve. These curves have been gotten by multiplying the density of states with the Fermi_Dirac distribution function, i.e. the density of electrons at a certain energy equals the density of states multiplied by the probability that the sate is occupied. n(E) = g(E) " f (E) . In the figure it is erroneously assumed that the Fermi level is the same for the two cases. We can see that this can not be the case because it will give a situation where the total number of electrons varies with temperature from the following argument. We call the total number of electrons pr volume for n and have " n= 0 ! " # n(E)dE = # g(E) f (E)dE . This is the shaded area or the area under the blue curve 0 and under the red curve in the two respective cases. These should be equal in order to conserve number of particles. As drawn they are not equal, which can be realized by observing that the difference between the two areas is equal to the differences between the two small (distorted triangular) areas A and B in the figure. It is obvious that the area A is larger than the area B. This means the Fermi level can not be constant with temperature. However if we make the Fermi level for the hot case, then we can conserve the number of electrons. Thus the Fermi level has to decrease with temperature. Argue from the above the Seebeck sign of the coefficient i.e give the sign of dV/dx when we have a positive temperature gradient. Her V is the electrostatic potential See sketch below It shows how the electron distributions, n(E) will vary along a sample with a temperature gradient in the x direction. From the above discussion we argued that the Fermi level decreases with temperature. Thus the distance from the Fermi-level to the bottom of the band becomes smaller. When there is no current flowing, the Fermi level is constant throughout the sample, thus the energy position of the bottom of the band varies thoughout the sample, which means there is a potential variation. The potential variation for electrons are drawn by the green line. In electricity the voltage = electrical potential is given for a positive particle, thus the hot end is at a lower electrostatic potential than the cold side. Above we thus have dT/dx>0 and dV/dx<0 Sketch haw the density of states in a system quantum wires is varying with energy. From the graph an in analogy with the idealized metal discussed above, argue the Seebeck coefficient now has changed sign. The density of states for quantum wires is given in the textbook and in the lecture notes. The typical feature is that dg/dE <0 in the 1D case whereas dg/dE>0 in the 3D case. The Seebeck coefficient is the change of potential/voltage with a temperature gradient across a sample. If we make a drawing as above with g(E) for the T=0 case and hot case on top of each other we have that the area B is smaller than the area A and we will have to increase EF with temperature in otder to satisfy the condition that no new particles are created or disappear.