FYS Vår 2014 (Kondenserte fasers fysikk)

FYS3410 - Vår 014 (Kondenserte fasers fysikk) http://www.uio.no/studier/emner/matnat/fys/fys3410/v14/index.html Pensum: Solid State Physics by Philip Hofmann (Chapters 1-7 and 11) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO Tel: +47-85776, e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen 3c

Lecture schedule (based on P.Hofmann s Solid State Physics,chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experimen h /1 Bragg planes and Brillouin zones (BZ) h 3/1 Elastic strain and structural defects h 3/1 Atomic diffusion and summary of Module I h Module II - Phonons 03/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models h 05/ Comparison of different models h 06/ Thermal conductivity h 07/ Thermal expansion and summary of Module II h Module III Electrons 4/ Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ FEFG in D and 1D, and DOS in nanostructures h 7/ Origin of the band gap and nearly free electron model h 8/ Number of orbitals in a band and general form of the electronic states h Module IV Semiconductors 10/3 Energy bands and effective mass method 4h 11/3 Density of states in 3D semiconductors and nanostructures h 1/3 Intrinsic semiconductors h 13/3 Impurity states in semiconductors and carrier statistics h 14/3 p-n junctions and optoelectronic devices h

Lecture 18: Energy bands in solids Bloch theorem Kronig-Penney model Empty lattice approximation Number of states in a band and filing of the bands Interpretation of the effective mass Effective mass method for hydrogen-like impurities

Lecture 18: Energy bands in solids Origin of the band gap and Bloch theorem Kronig-Penney model Empty lattice approximation Number of states in a band and filing of the bands Interpretation of the effective mass Effective mass method for hydrogen-like impurities

Why do we get a gap? Let us start with a free electron in a periodic crystal, but ignore the atomic potentials for now At the interface (BZ), we have two counter-propagating waves e ikx, with k = p/a, that Bragg reflect and form standing waves y E Its periodically extended partner -p/a p/a k

Why do we get a gap? y + ~ cos(px/a) peaks at atomic sites y - ~ sin(px/a) peaks in between y + y - E Its periodically extended partner -p/a p/a k

Let s now turn on the atomic potential The y + solution sees the atomic potential and increases its energy The y - solution does not see this potential (as it lies between atoms) Thus their energies separate and a gap appears at the BZ This happens only at the BZ where we have standing waves U 0 y + y - -p/a p/a k

Bloch theorem If V(r) is periodic with the periodicity of the lattice, then the solutions of the one-electron Schrödinger eq. Where u k (r) is periodic with the periodicity of the direct lattice. u k (r) = u k (r+t); T is the translation vector of lattice. The eigenfunctions of the wave equation for a periodic potential are the product of a plane wave exp(ik r) times a function u k (r) with the periodicity of the crystal lattice.

Kronig-Penney Model Kronig and Penney assumed that an electron experiences an infinite one-dimensional array of finite potential wells. Each potential well models attraction to an atom in the lattice, so the size of the wells must correspond roughly to the lattice spacing.

Kronig-Penney Model An effective way to understand the energy gap in semiconductors is to model the interaction between the electrons and the lattice of atoms. R. de L. Kronig and W. G. Penney developed a useful one-dimensional model of the electron lattice interaction in 1931.

Kronig-Penney Model Since the electrons are not free their energies are less than the height V 0 of each of the potentials, but the electron is essentially free in the gap 0 < x < a, where it has a wave function of the form where the wave number k is given by the usual relation:

Kronig-Penney Model In the region between a < x < a + b the electron can tunnel through and the wave function loses its oscillatory solution and becomes exponential:

Kronig-Penney Model The left-hand side is limited to values between +1 and 1 for all values of K. Plotting this it is observed there exist restricted (shaded) forbidden zones for solutions.

Kronig-Penney Model Matching solutions at the boundary, Kronig and Penney find Here K is another wave number. Let s label the equation above as KPE Kronig-Penney equation

Kronig-Penney Model (a) Plot of the left side of Equation (KPE) versus ka for κ ba / = 3π /. Allowed energy values must correspond to the values of k for for which the plotted function lies between -1 and +1. Forbidden values are shaded in light blue. (b) The corresponding plot of energy versus ka for κ ba / = 3π /, showing the forbidden energy zones (gaps).

Empty lattice approximation Suppose that we have empty lattice where the periodic V(x)=0. Then the e - s in the lattice are free, so that

Empty lattice approximation in 3-dim.

Empty lattice approximation In simple cubic (SC) empty lattice

Empty lattice approximation Everything can be described within the 1 st B.Z.

E E Number of states in a band k np d Sin k p o d Sin90 k p o d Sin 45 [100] [110] Effective gap p d k p d k

Filing of the bands Divalent metals Monovalent metals Monovalent metals: Ag, Cu, Au 1 e in the outermost orbital outermost energy band is only half filled Divalent metals: Mg, Be overlapping conduction and valence bands they conduct even if the valence band is full Trivalent metals: Al similar to monovalent metals!!! outermost energy band is only half filled!!!

Filing of the bands SEMICONDUCTORS and ISOLATORS Band gap Elements of the 4 th column (C, Si, Ge, Sn, Pb) valence band full but no overlap of valence and conduction bands Diamond PE as strong function of the position in the crystal Band gap is 5.4 ev Down the 4 th column the outermost orbital is farther away from the nucleus and less bound the electron is less strong a function of the position in the crystal reducing band gap down the column

Multiple Quantum Wells (MQWs) for advanced LEDs Single quantum well - repetitions of ZnO/ZnCdO/ZnO periods MQWs

Internal and external forces affecting an electron in crystal The electron is subject to internal forces from the lattice (ions and core electrons) AND external forces such as electric fields In a crystal lattice, the net force may be opposite the external force, however: - F ext =-qe F int =-de p /dx E p (x) + + + + +

Internal and external forces affecting an electron in crystal electron acceleration is not equal to F ext /m e, but rather a = (F ext + F int )/m e == F ext /m * The dispersion relation E(K) compensates for the internal forces due to the crystal and allows us to use classical concepts for the electron as long as its mass is taken as m * - F ext =-qe F int =-de p /dx E p (x) + + + + +

Hole - an electron near the top of an energy band The hole can be understood as an electron with negative effective mass An electron near the top of an energy band will have a negative effective mass A negatively charged particle with a negative mass will be accelerated like a positive particle with a positive mass (a hole!) E(K) p/a K F = m * a = QE Without the crystal lattice, the hole would not exist! The hole is a pure consequence of the periodic potential operating in the crystal!!!

E(K) and E(x) E(K) E(x) E C - conduction band K E V + valence band E g x p/a

Generation and Recombination of electron-hole pairs E(x) conduction band E C - - E V + valence band + x

Generation and Recombination of electron-hole pairs Single quantum well - repetitions of ZnO/ZnCdO/ZnO periods MQWs

Real 3D lattices, e.g. FCC, BCC, diamond, etc. a E(K x ) E(K y ) b y x p/a K x p/b K y Different lattice spacings lead to different curvatures for E(K) and effective masses that depend on the direction of motion.

Real 3D lattices, e.g. FCC, BCC, diamond, etc. m 1 1 c, ij E kk i j heavy m * (smaller d E/dK ) light m * (larger d E/dK )

Real 3D lattices, e.g. FCC, BCC, diamond, etc. Ge Si GaAs

Direct and inderect band gap in semiconductors energy (E) and momentum (ħk) must be conserved energy is released when a quasi-free electron recombines with a hole in the valence band: ΔE = E g does this energy produce light (photon) or heat (phonon)? indirect bandgap: ΔK is large but for a direct bandgap: ΔK=0 photons have very low momentum but lattice vibrations (heat, phonons) have large momentum Conclusion: recombination (e - +h + ) creates light in direct bandgap materials (GaAs, GaN, etc) heat in indirect bandgap materials (Si, Ge)

Motion of free electrons Consider free electron E(p) Dispersion relation E p / m =( p + p + p ) / m pm v 0 x y z 0 0 Newton s law of motion Velocity Mass 0 dp p F x dt m 0 dr p dt p E E E E E v v= v xˆ v yˆ v zˆ xˆ yˆ zˆ p x x x y z m0 px px py pz E E E m 1 0 = px py pz

E Effective mass Taylor series near CB minimum 3 1 E E( k) E ( k k )( k k )... g i 0i j 0j i, j1 ki kj k ox k x k oy C.B. minimum k y Effective mass tensor k 0 3 1 E E( k) E ( k k )( k k )... g i 0i j 0 j i, j1kikj 1 E m p p... p ( k k ) 3 1 g c, ij i j i i 0i i, j1 m 1 1 c, ij E kk i j

Density of states k z Parallelepiped of Volume V=L x L y L z Allowed states satisfy boundary conditions: k L pn i i i k y k x Each state occupies volume in k space V 8 p /( L L L ) 8p V 3 3 1 k x y z Consider a conduction valley along x or [100] direction k E( k) Eg m m m kx y kz L T T k k x y kz 1 m ( E E ) m ( E E ) m ( E E ) L g T g T g Half axes m ( E E ) m ( E E ) m ( E E ) a, a =, a = ; L g T g T g x y z The volume in k-space containing energies less than E 3/ 4 4 1/ 3/ k ( ) p x y z p ( ) L T g V E a a a m m E E 3 3

Density of states k z 3/ 4 1/ 3/ k ( ) p ( ) L T g V E m m E E 3 k x Number of states with energies less than E k y 3/ V 1/ 3/ k k L T c N( E) V ( E) / V m m ( E E ) 3p Multiply by number of valleys -G, divide by volume take derivative 3/ * 3/ N( E) 1 G 1/ 1/ 1 m n 1/ c( ) ( ) ( ) L T g g g E G V m m E E E E E p p Number of states per unit volume per unit energy density of states Effective D.O.S. mass m * G /3 m m 1/3 n L T In GaAs m * n m c

Density of states in the valence band k Count energy down E 3/ 3/ * 3/ * m lh 1/ m m hh 1/ p 1/ 1 1 1 gv( E) glh( E) ghh( E) E E E p p p * 3/ 3/ Effective D.O.S. mass is /3 m m m ~ m p hh lh hh

Lecture 18: Energy bands in periodic crystalls Origin of the band gap and Bloch theorem Kronig-Penney model Empty lattice approximation Number of states in a band and filing of the bands Interpretation of the effective mass Effective mass method for hydrogen-like impurities in semiconductors

Hydrogen like impurities in semiconductors P donor in Si can be modeled d as hydrogen-like atom Hydrogen atom Hydrogen-like donor

Hydrogen atom - Bohr model 0 4 0 0 0 0 0 0 0 0 0 0 1 ) (4 ) (4 energy: Total 4 1 Kineticenergy: 4 4 Potentialenergy: 4 1 4 4 ) ( 4 4 1,,3..., for 4 1 n e mz E K r Ze V K E r Ze mv K r Ze dr r Ze V n Ze mr n v mze n r mr n mr n mr r mv Ze n n mvr L r v m r Ze r p p p p p p p p p p p

Hydrogen atom - Bohr model E 4 m0 q H. (4p0) 13 6 ev

Hydrogen like impurities in semiconductors Hydrogen-like donor Instead of m 0, we have to use m n*. Instead of o, we have to use K s o. K s is the relative dielectric constant of Si (K s, Si = 11.8). E 4 m0 q H. (4p0) 13 6 ev E d m * n 4 q (4π K ) s 0 13. 6 ev m * n m0 0 Ks0 0.05 ev

Lecture schedule (based on P.Hofmann s Solid State Physics,chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experimen h /1 Bragg planes and Brillouin zones (BZ) h 3/1 Elastic strain and structural defects h 3/1 Atomic diffusion and summary of Module I h Module II - Phonons 03/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models h 05/ Comparison of different models h 06/ Thermal conductivity h 07/ Thermal expansion and summary of Module II h Module III Electrons 4/ Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ FEFG in D and 1D, and DOS in nanostructures h 7/ Origin of the band gap and nearly free electron model h 8/ Number of orbitals in a band and general form of the electronic states h Module IV Semiconductors 10/3 Energy bands and effective mass method 4h 11/3 Energy band structure (continuation) h 1/3 Intrinsic semiconductors h 13/3 Impurity states in semiconductors and carrier statistics h 14/3 p-n junctions and optoelectronic devices h

Lecture 19: Energy band structure (continuation) Dynamics of electrons in a band Band-to-band transitions

Dynamics of electrons in a band The external electric field causes a change in the k vectors of all electrons: dk d k e E F ee dt dt E If the electrons are in a partially filled band, this will break the symmetry of electron states in the 1 st BZ and produce a net current. But if they are in a filled band, even though all electrons change k vectors, the symmetry remains, so J = 0. p a v p a k x k x When an electron reaches the 1 st BZ edge (at k = p/a) it immediately reappears at the opposite edge (k = -p/a) and continues to increase its k value. As an electron s k value increases, its velocity increases, then decreases to zero and then becomes negative when it re-emerges at k = -p/a!!

Dynamics of electrons in a band

Band-to-band transitions 5

Band-to-band transitions valence band conduction band

Band-to-band transitions valence band conduction band gap size (ev) InSb 0.18 InAs 0.36 Ge 0.67 Si 1.11 GaAs 1.43 SiC.3 diamond 5.5 MgF 11

Band-to-band transitions valence band conduction band

Band-to-band transitions electrons in the conduction band (CB) missing electrons (holes) in the valence band (VB)

Band-to-band transitions

Band-to-band transitions free electrons VB maximum as E=0 conduction band valence band

Band-to-band transitions electrons in the conduction band (CB) missing electrons (holes) in the valence band (VB)

Band-to-band transitions for the conduction band for the valence band Both are Boltzmann distributions! This is called the non-degenerate case.

Band-to-band transitions

Band-to-band transitions CBM μ VBM

Band-to-band transitions In quantum wells

Lecture schedule (based on P.Hofmann s Solid State Physics,chapters 1-7 and 11) Module I Periodic Structures and Defects 0/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space 4h 1/1 Laue condition, Ewald construction, interpretation of a diffraction experimen h /1 Bragg planes and Brillouin zones (BZ) h 3/1 Elastic strain and structural defects h 3/1 Atomic diffusion and summary of Module I h Module II - Phonons 03/ Vibrations, phonons, density of states, and Planck distribution 4h 04/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models h 05/ Comparison of different models h 06/ Thermal conductivity h 07/ Thermal expansion and summary of Module II h Module III Electrons 4/ Free electron gas (FEG) versus Free electron Fermi gas (FEFG) 4h 5/ Effect of temperature Fermi- Dirac distribution h 6/ FEFG in D and 1D, and DOS in nanostructures h 7/ Origin of the band gap and nearly free electron model h 8/ Number of orbitals in a band and general form of the electronic states h Module IV Semiconductors 10/3 Energy bands and effective mass method 4h 11/3 Energy bands (continuation) h 1/3 Intrinsic semiconductors h 13/3 Impurity states in semiconductors and carrier statistics h 14/3 p-n junctions and optoelectronic devices h

Lecture 1: Impurity states in semiconductors and carrier statistics Intrinsic and extrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

Intrinsic and extrinsic semiconductors

Intrinsic and extrinsic semiconductors E g n i E i E g n 0 E i E g n 0 E i p i p 0 p 0

Intrinsic and extrinsic semiconductors n 0 E g E ID n d E d E i p 0

Hydrogen like impurities in semiconductors d P donor in Si can be modeled as hydrogen-like atom Hydrogen atom Hydrogen-like donor

Hydrogen atom - Bohr model E 4 m0 q H. (4p0) 13 6 ev

Hydrogen like impurities in semiconductors n 0 =0 n 0 E g E ID E g E ID E i E d n d E i E d p 0 =0 p 0

N- and p-type semiconductors Intrinsic semiconductor a) Energy level diagrams showing the excitation of an electron from the valence band to the conduction band. The resultant free electron can freely move under the application of electric field. b) Equal electron & hole concentrations in an intrinsic semiconductor created by the thermal excitation of electrons across the band gap

N- and p-type semiconductors n-type Semiconductor a) Donor level in an n-type semiconductor. b) The ionization of donor impurities creates an increased electron concentration distribution.

N- and p-type semiconductors p-type Semiconductor a) Acceptor level in an p-type semiconductor. b) The ionization of acceptor impurities creates an increased hole concentration distribution

N- and p-type semiconductors

N- and p-type semiconductors Intrinsic material: A perfect material with no impurities. n p n n& p & n i i E exp( k g B ) T are the electron, hole & intrinsic concentrations respectively. E g is the gap energy, T is Temperature. Extrinsic material: donor or acceptor type semiconductors. pn n i Majority carriers: electrons in n-type or holes in p-type. Minority carriers: holes in n-type or electrons in p-type.

N- and p-type semiconductors donor: impurity atom that increases n acceptor: impurity atom that increases p N-type material: contains more electrons than holes P-type material: contains more holes than electrons majority carrier: the most abundant carrier minority carrier: the least abundant carrier intrinsic semiconductor: n = p = n i extrinsic semiconductor: doped semiconductor

Equilibrium charge carrier concentration in semiconductors Consider conditions for charge neutrality. The net charge in a small portion of a uniformly doped semiconductor should be zero. Otherwise, there will be a net flow of charge from one point to another resulting in current flow (that is against out assumption of thermal equilibrium). Charge/cm 3 = q p q n + q N D + q N A = 0 or p n + N D + N A = 0 where N D + = # of ionized donors/cm 3 and N A = # of ionized acceptors per cm 3. Assuming total ionization of dopants, we can write:

Equilibrium charge carrier concentration in semiconductors Assume a non-degenerately doped semiconductor and assume total ionization of dopants. Then, n p = n i ; electron concentration hole concentration = n i p n + N D N A = 0; net charge in a given volume is zero. Solve for n and p in terms of N D and N A We get: (n i / n) n + N D N A = 0 n n (N D N A ) n i = 0 Solve this quadratic equation for the free electron concentration, n. From n p = n i equation, calculate free hole concentration, p.

Equilibrium charge carrier concentration in semiconductors Intrinsic semiconductor: N D = 0 and N A = 0 p = n = n i Doped semiconductors where N D N A >> n i n = N D N A ; p = n i / n if N D > N A p = N A N D ; n = n i / p if N A > N D Compensated semiconductor n = p = n i when n i >> N D N A When N D N A is comparable to n i,, we need to use the charge neutrality equation to determine n and p.

Equilibrium charge carrier concentration in semiconductors Example Si is doped with 10 17 As Atom/cm 3. What is the equilibrium hole concentra-tion p 0 at 300 K? Where is E F relative to E i ni p0 10 n 10 0 0.510 3 3.5 cm 17 n 0 n e i ( E F E i ) kt 17 n 10 EF Ei ktln 0 0.059ln 0. 407eV 10 n 1.510 i

Equilibrium charge carrier concentration in semiconductors n N p N 0 a 0 d

Charge carriers in non-eqilibrium conditions Particles diffuse from regions of higher concentration to regions of lower concentration region, due to random thermal motion.

Charge carriers in non-eqilibrium conditions J n,diff qd n dn dx J p,diff qd p dp dx D is the diffusion constant, or diffusivity.

Charge carriers in non-eqilibrium conditions J J J n p J n J n, drift J n, diff qn ε n qd n dn dx J p J p, drift J p, diff qp ε p qd p dp dx

Charge carriers in non-eqilibrium conditions The position of E F relative to the band edges is determined by the carrier concentrations, which is determined by the net dopant concentration. In equilibrium E F is constant; therefore, the band-edge energies vary with position in a non-uniformly doped semiconductor: E c (x) E F E v (x)

The ratio of carrier densities at two points depends exponentially on the potential difference between these points: 1 i i1 1 1 i 1 i i i1 i F i i 1 F i1 i 1 i1 F ln 1 ln ln ln Therefore ln Similarly, ln ln n n q kt E E q V V n n kt n n n n kt E E n n kt E E n n kt E E n n kt E E Charge carriers in non-eqilibrium conditions

E v (x) Charge carriers in non-eqilibrium conditions E f E c (x) Consider a piece of a non-uniformly doped semiconductor: n-type semiconductor Decreasing donor concentration E c (x) E F E v (x) n dn dx Nc e kt N e c n kt n kt ( E E ( Ec EF ) / kt de dx c qε c F )/ kt de dx c

Charge carriers in non-eqilibrium conditions If the dopant concentration profile varies gradually with position, then the majority-carrier concentration distribution does not differ much from the dopant concentration distribution. N x) p( x) N ( x) n( x) D( A n-type material: p-type material: n( x) ND( x) NA p( x) NA( x) ND ( x) ( x) kt q 1 n dn dx kt q 1 N D dn dx D in n-type material

Charge carriers in non-eqilibrium conditions Band-to-Band R-G Center Impact Ionization

Charge carriers in non-eqilibrium conditions Direct R-G Center Auger

Charge carriers in non-eqilibrium conditions Energy (E) vs. momentum (ħk) Diagrams Direct: Indirect: Little change in momentum is required for recombination momentum is conserved by photon emission Large change in momentum is required for recombination momentum is conserved by phonon + photon emission

Charge carriers in non-eqilibrium conditions equilibrium values n n n 0 p p p 0 Charge neutrality condition: n p

Charge carriers in non-eqilibrium conditions Often the disturbance from equilibrium is small, such that the majority-carrier concentration is not affected significantly: For an n-type material: n p n0 so n n0 For a p-type material: n p p0 so p p0 However, the minority carrier concentration can be significantly affected.

Charge carriers in non-eqilibrium conditions Consider a semiconductor with no current flow in which thermal equilibrium is disturbed by the sudden creation of excess holes and electrons. n t n n for electrons in p-type material p t p p for holes in n-type material

Charge carriers in non-eqilibrium conditions Uniformly doped p-type and n- type semiconductors before the junction is formed. Internal electric-field occurs in a depletion region of a p-n junction in thermal equilibrium