Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 1 Chap 3. Introduction to Quantum Theory of Solids  Allowed and Forbidden Energy Bands  k-space Diagrams  Electrical Conduction in Solids  Density of State Functions  Statistical Mechanics  Homework

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 2 Preview  Recall from the previous analysis that the energy of a bound electron is quantized. And for the one-electron atom, the probability of finding the electron at a particular distance from the nucleus is not localized at a given radius.  Consider two atoms that are in close proximity to each other. The wave functions of the two atom electrons overlap, which means that the two electrons will interact. This interaction results in the discrete quantized energy level splitting into two discrete energy levels.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 3 Formation of Energy Bands  Consider a regular periodic arrangement of atoms in which each atoms contains more than one electron. If the atoms are initially far apart, the electrons in adjacent atoms will not interact and will occupy the discrete energy levels.  If the atoms are brought closer enough, the outmost electrons will interact and the energy levels will split into a band of allowed energies.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 4 Formation of Energy Bands

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 5 Kronig-Penny Model  The concept of allowed and forbidden energy levels can be developed by considering Schrodinger’s equation. Kronig-Penny Model

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 6 Kronig-Penny Model  The Kronig-Penny model is an idealized periodic potential representing a 1-D single crystal.  We need to solve Schrodinger’s equation in each region.  To obtain the solution to the Schrodinger’s equation, we make use of Bloch theorem. Bloch states that all one-electron wave functions, involving periodically varying potential energy functions, must be of the form,  (x) = u(x)e jkx, u(x) is a periodic function with period (a+b) and k is called a constant of the motion.  The total wave function  (x,t) may be written as  (x,t) = u(x)e j(kx-(E/ħ)t).  In region I (0 < x < a), V(x) = 0, then Schrodinger’s equation becomes

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 7 Kronig-Penny Model  The solution in region I is of the form,  In region II (-b < x < 0), V(x) = V o, and apply Schrodinger’s eq. The solution for region II is of the form,  Boundary conditions:

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 8 Kronig-Penny Model  There is a nontrivial solution if, and only if, the determinant of the coefficients is zero. This result is  The above equation relates k to the total energy E (through  ) and the potential function V o (through  ). The allowed values of E can be determined by graphical or numerical methods.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 9 Kronig-Penny Model  Recall -1  cosk(a+b)  1, so E-values which cause f(  ) to lie in the range -1  f(  )  1 are the allowed system energies.—  The ranges of allowed energies are called energy bands; the excluded energy ranges (|f(  )|  1) are called the forbidden gaps or bandgaps.  The energy bands in a crystal can be visualized by  Energy

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 10 E-k Diagram

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 11 k-space Diagram  Consider the special case for which V o = 0, (free particle case)  cos  (a+b) = cosk(a+b), i.e.,  = k,,where p is the particle momentum and k is referred as a wave number.  We can also relate the energy and momentum as E = k 2 ħ 2 /2m

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 12 E-k diagram  More interesting solution occur for E < V o (  = j  ), which applies to the electron bound within the crystal. The result could be written as  Consider a special case, b  0, V o , but bV o is finite, the above eq. becomes  The solution of the above equation results in a band of allowed energies.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 13 E-k diagram  Consider the function ofgraphically,

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 14 E-k diagram  E-k diagram could be generated from the above figure.  This shows the concept of the allowed energy bands for the particle propagating in the crystal.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 15 Reduced k-space

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 16 Electrical Conduction in Solids  the Bond Model  Energy Band E-K diagram of a semiconductor

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 17 Drift Current  If an external force is applied to the electrons in the conduction band and there are empty energy states into which the electrons can move, electrons can gain energy and a net momentum.  The drift current due to the motion of electrons is where n is the number of electrons per volume and v i is the electron velocity in the crystal.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 18 Electron Effective Mass  The movement of an electron in a lattice will be different than that of an electron in free space. There are internal forces in the crystal due to the positively charged ions or protons and electrons, which will influence the motion of electrons in the crystal. We can write  Since it is difficult to take into account of all of the internal forces, we can write  m * is called the effective mass which takes into account the particle mass and the effect of the internal forces.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 19 Effective mass, E-k diagram  Recall for a free electron, the energy and momentum are related by –So the first derivative of E w.r.t. k is related to the velocity of the particle.  In addition, –So the second derivative of E w.r.t. k is inversely proportional to the mass of the particle.  In general, the effective mass could be related to

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 20 Effective mass, E-k diagram  m* >0 near the bottoms of all band; m* <0 near the tops of all bands  m* <0 means that, in response to an applied force, the electron will accelerate in a direction opposite to that expected from purely classical consideration.  In general, carriers are populated near the top or bottom band edge in a semiconductor—the E-k relationship is typically parabolic and, therefore, thus carriers with energies near the top or bottom of an energy band typically exhibit a CONSTANT effective mass

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 21 Concept of Hole

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 22 Extrapolation of Concepts to 3-D  Brilliouin Zones: is defined as a Wigner-Seitz cell in the reciprocal lattice.   point: Zone center (k = 0)  (0 0 0 )  X point: Zone-boundary along a direction  6 symmetric points (1 0 0) (-1 0 0) (0 1 0) (0 -1 0) (0 0 1) (0 0 -1)  L point: Zone-boundary along a direction  8 symmetric points  , X, and L points are highly symmetric  energy stable states  carriers accumulate near these points in the k-space.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 23 E-k diagram of Si, Ge, GaAs

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 24 Energy Band  Valence Band: –In all cases the valence-band maximum occurs at the zone center, at k = 0 –is actually composed of three subbands. Two are degenerate at k = 0, while the third band maximizes at a slightly reduced energy. The k = 0 degenerate band with the smaller curvature about k = 0 is called “heavy-hole” band, and the k = 0 degenerate band with the larger curvature is called “light-hole” band. The subband maximizing at a slightly reduced energy is the “split-off” band. –Near k = 0 the shape and the curvature of the subbands is essentially orientation independent.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 25 Energy Band  Conduction band: –is composed of a number of subbands. The various subbands exhibit localized and abssolute minima at the zone center or along one of the high-symmetry diirections. –In Ge the conduction-band minimum occurs right at the zone boundary along direction. ( there are 8 equivalent conduction-band minima.) –The Si conduction-band minimum occurs at k~0.9(2  /a) from the zone center along direction. (6 equivalent conduction-band minima) –GaAs has the conduction-band minimum at the zone center directly over the valence-band maximum. Morever, the L-valley at the zone boundary direction lies only 0.29 eV above the conduction-band minimum. Even under equilibrium, the L-valley contains a non-negligible electron population at elevated temp. The intervalley transition should be taken into account.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 26 Metal, Semiconductor, and Insulator Insulator Semiconductor Metal

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 27 The k-space of Si and GaAs  Direct bandgap: the valence band maximum and the conduction band minimum both occur at k = 0. Therefore, the transition between the two allowed bands can take place without change in crystal momentum.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 28 Constant-Energy Surfaces  A 3-D k-space plot of all the allowed k-values associated with a given energy E. The geometrical shapes, being associated with a given energy, are called constant-energy surfaces (CES).  Consider the CES’s characterizing the conduction-band structures near E c in Ge, Si, and GaAs. (a) Constant-energy surfaces(b) Ge surface at the Brillouin-zone boundaries.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 29 Constant-Energy Surfaces of E c  For Ge, Ec occurs along each of the 8 equivalent directions; a Si conduction band minimum, along each of 6 equivalent directions. For GaAs, E c is positioned at the zone center, giving rise to a single constant-energy surface.  For energy slightly removed from E c : E-E c  Ak 1 2 +Bk 2 2 +Ck 3 2, where k 1, k 2, k 3 are k-space coordinates measured from the center of a band minimum along principle axes. For example: Ge, the k 1, k 2, k 3 coordinate system would be centered at the [111] L-point and one of the coordinate axes, say k 1 -axis, would be directed along the k x -k y -k z [111] direction.  For GaAs, A = B = C, exhibits spherical CES;  For Ge and Si, B=C, the CES’s are ellipsoids of revolution.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 30 Effective Mass  In 3-D crystals the electron acceleration arising from an applied force is analogously by where  For GaAs,, so m ij = 0 if i  j, and therefore, we can define m ii =m e *, that is the the effective mass tensor reduces to a scalar, giving rise to an orientation-indep. equation of motion like that of a classical particle.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 31 Effective Mass  For Si and Ge: E-E c = Ak 1 2 +B(k 2 2 +k 3 2 ) so m ij = 0 if i  j, and  Because m 11 is associated with the k-space direction lying along the axis of revolution, it is called the longitudinal effective mass m l *. Similarly, m 22 = m 33, being associated with a direction perpendicular to the axis of revolution, is called the transverse effective mass m t *.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 32 Effective Mass  The relative sizes of m l * and m t * can be deduced by inspection of the Si and Ge constant-energy plots.  For both Ge and Si, m l * > m t *. Further, m l * /m t * of Ge > m l * /m t * of Si.  The valence-band structure of Si, Ge, and GaAs are approximately spherical and composed of three subbands. Thus, the holes in a given subband can be characterized by a single effective mass parameter, but three effective mass (m hh *, m lh *, and m so * ) are required to characterize the entire hole population. The split-off band, being depressed in energy, is only sparsely populated and is often ignored.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 33 Effective Mass measurement  The near-extrema point band structure, multiplicity and orientation of band minima, etc. were all originally confirmed by cyclotron resonance measurement.  Resonance experiment is performed in a microwave resonance cavity at temperature 4K. A static B field and an rf E-field oriented normal to B are applied across the sample. The carriers in the sample will move in an orbit-like path about the direction of B and the cyclotron frequency  c = qB/mc. When the B-field strength is adjusted such that  c = the  of the rf E-field, the carriers absorb energy from the E-field (in resonance).  m= qB/  c  Repeating the different B-field orientations allows one to separate out the effective mass factors (m l * and m t *)

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 34 Effective Mass of Si, Ge, and GaAs

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 35 Density of State Function  To calculate the electron and hole concentrations in a material, we must determine the density of these allowed energy states as a function of energy.  Electrons are allowed to move relatively freely in the conduction band of a semiconductor but are confined to the crystal.  To simulate the density of allowed states, consider an appropriate model: A free electron confined to a 3-D infinite potential well, where the potential well represents the crystal.  The potential of the well is defined as V(x,y,z) = 0 for 0<x<a, 0<y<a, 0<z<a, and V(x,y,z) =  elsewhere  Solving the Schrodinger’s equation, we can obtain

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 36 Density of State Function   The volume of a single quantum state is V k =(  /a) 3, and the differential volume in k-space is 4  k 2 dk  Therefore, we can determine the density of quantum states in k-space as –The factor, 2, takes into account the two spin states allowed for each quantum state; the next factor, 1/8, takes into account that we are considering only the quantum states for positive values of k x, k y, and k z.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 37 Density of State Function  Recall that  We can determine the density of states as a function of energy E by  Therefore, the density of states per unit volume is given by  Extension to semiconductors, the density of states in conduction band is modified as and the density of states in valence band is modified as

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 38 Density of State Function  m n * and m p * are the electron and hole density of states effective masses. In general, the effective mass used in the density of states expression must be an average of the band-structure effective masses.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 39 Density of States Effective Mass  Conduction Band--GaAs: the GaAs conduction band structure is approximately spherical and the electronss within the band are characterized by a single isotropic effective mass, m e *,   Conduction Band--Si, Ge: the conduction band structure in Si and Ge is characterized by ellipsoidal energy surfaces centered, respectively, at points along the and directions in k-space.  Valence Band--Si, Ge, GaAs: the valence band structures are al characterized by approximately spherical constant-energy surfaces (degenerate).

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 40 Density of States Effective Mass

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 41 Statistics Mechanics  In dealing with large numbers of particles, we are interested only in the statistical behavior of the whole group rather than in the behavior of each individual particle.  There are three distribution laws determining the distribution of particles among available energy states.  Maxwell-Boltzmann probability function: –Particles are considered to be distinguishable by being numbered for 1 to N with no limit to the number of particles allowed in each energy state.  Bose-Einstein probability function: –Particles are considered to be indistinguishable and there is no limit to the number of particles permitted in each quantum state. (e.g., photons)  Fermi-Dirac probability function: –Particles are indistinguishable but only one particle is permitted in each quantum state. (e.g., electrons in a crystal)

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 42 Fermi-Dirac Distribution  Fermi-Dirac distribution function gives the probability that a quantum state at the energy E will be occupied by an electron.  the Fermi energy (E F ) determine the statistical distribution of electrons and does not have to correspond to an allowed energy level.  At T = 0K, f(E E F ) = 0, electrons are in the lowest possible energy states so that all states below E F are filled and all states above E F are empty.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 43 Fermi-Dirac Distribution, at T=0K

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 44 Fermi-Dirac Distribution  For T > 0K, electrons gain a certain amount of thermal energy so that some electrons can jump to higher energy levels, which means that the distribution of electrons among the available energy states will change.  For T > 0K, f(E = E F ) = ½

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 45 Boltamann Approximation  Consider T >> 0K, the Fermi-Dirac function could be approximated by which is known as the Maxwell-Boltzmann approximation.

Solid-State Electronics Chap. 3 Instructor: Pei-Wen Li Dept. of E. E. NCU 46 Homework  3.5  3.8  3.16