FYS Vår 2015 (Kondenserte fasers fysikk)

Transcription

1 FYS Vår 015 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 0, and Appendix D) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen 3a

2 FYS3410 Lectures (based on C.Kittel s Introduction to SSP, Chapters 1-9, 17,18,0) Module I Periodic Structures and Defects (Chapters 1-3, 0) 6/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space h 7/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 8/1 Elastic strain and structural defects in crystals h 30/1 Atomic diffusion and summary of Module I h Module II Phonons (Chapters 4 and 5) 09/ Vibrations, phonons, density of states, and Planck distribution h 10/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models. Planck distribution. Comparison of different models 4h 11/ Thermal conductivity h 13/ Thermal expansion and summary of Module II. h Module III Electrons (Chapters 6, 7, 18 - pp , and Appendix D) 3/ Free electron gas (FEG) versus free electron Fermi gas (FEFG) h 4/ Effect of temperature Fermi- Dirac distribution. Heat capacity of FEFG. DOS in nanostructures. Origin of the band gap 4h 5/ Nearly free electron model. Kronig-Penney model. Empty lattice approximation. h 7/ Number of orbitals in a band. Summary of Module III. h Module IV Semiconductors and interfaces (Chapters 8, 9-pp 3-31, 17) 09/3 Recap of energy bands. Metals versus semiconductors. Surfaces and interfaces. h 10/3 Effective mass method. Intrinsic carrier generation elctrons and holes. Localized levels for hydrogen-like impurities donors and acceptors. Doping. 4h 11/3 Carrier statistics in semiconductors. h 13/3 p-n junctions. Examples of optoelectronic devices. Summary of Module IV. h

3 Lecture 13: Recap of energy bands. Metals versus semiconductors. Surfaces and interfaces. Introduction to the effective mass method Recap of energy bands Metals versus semiconductors Surfaces and interfaces Introduction to the effective mass method

4 Lecture 13: Recap of energy bands. Metals versus semiconductors. Surfaces and interfaces. Introduction to the effective mass method Recap of energy bands Metals versus semiconductors Surfaces and interfaces Introduction to the effective mass method

5 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 -p/a p/a k

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

7 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

8 Lecture 13: Recap of energy bands. Metals versus semiconductors. Surfaces and interfaces. Introduction to the effective mass method Recap of energy bands Metals versus semiconductors Surfaces and interfaces Introduction to the effective mass method

9

10

11 Lecture 13: Recap of energy bands. Metals versus semiconductors. Surfaces and interfaces. Introduction to the effective mass method Recap of energy bands Metals versus semiconductors Surfaces and interfaces Introduction to the effective mass method

12

13

14

15

16

17

18

19

20 Lecture 13: Recap of energy bands. Metals versus semiconductors. Surfaces and interfaces. Introduction to the effective mass method Recap of energy bands Metals versus semiconductors Surfaces and interfaces Introduction to the effective mass method

21

22

23 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 with a negative curvature 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) F = m * a = QE p/a K Without the crystal lattice, the hole would not exist! The hole is a pure consequence of the periodic potential operating in the crystal!!!

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

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

26 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.

27 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 )

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

29 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)

30 FYS Vår 015 (Kondenserte fasers fysikk) Pensum: Introduction to Solid State Physics by Charles Kittel (Chapters 1-9 and 17, 18, 0, and Appendix D) Andrej Kuznetsov delivery address: Department of Physics, PB 1048 Blindern, 0316 OSLO Tel: , e-post: andrej.kuznetsov@fys.uio.no visiting address: MiNaLab, Gaustadaleen 3a

31 FYS3410 Lectures (based on C.Kittel s Introduction to SSP, Chapters 1-9, 17,18,0) Module I Periodic Structures and Defects (Chapters 1-3, 0) 6/1 Introduction. Crystal bonding. Periodicity and lattices, reciprocal space h 7/1 Laue condition, Ewald construction, interpretation of a diffraction experiment Bragg planes and Brillouin zones 4h 8/1 Elastic strain and structural defects in crystals h 30/1 Atomic diffusion and summary of Module I h Module II Phonons (Chapters 4 and 5) 09/ Vibrations, phonons, density of states, and Planck distribution h 10/ Lattice heat capacity: Dulong-Petit, Einstien and Debye models. Planck distribution. Comparison of different models 4h 11/ Thermal conductivity h 13/ Thermal expansion and summary of Module II. h Module III Electrons (Chapters 6, 7, 18 - pp , and Appendix D) 3/ Free electron gas (FEG) versus free electron Fermi gas (FEFG) h 4/ Effect of temperature Fermi- Dirac distribution. Heat capacity of FEFG. DOS in nanostructures. Origin of the band gap 4h 5/ Nearly free electron model. Kronig-Penney model. Empty lattice approximation. h 7/ Number of orbitals in a band. Summary of Module III. h Module IV Semiconductors and interfaces (Chapters 8, 9-pp 3-31, 17) 09/3 Recap of energy bands. Metals versus semiconductors. Surfaces and interfaces. h 10/3 Effective mass method. Intrinsic carrier generation elctrons and holes. Localized levels for hydrogen-like impurities donors and acceptors. Doping. 4h 11/3 Carrier statistics in semiconductors. h 13/3 p-n junctions. Examples of optoelectronic devices. Summary of Module IV. h

32 Lecture 14: Intrinsic and extrinsic semiconductors recap of the effective mass method analysis of E(k) and it s 1st and nd deriviates band-to-band transiotions: intrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

33 Lecture 14: Intrinsic and extrinsic semiconductors recap of the effective mass method analysis of E(k) and it s 1st and nd deriviates band-to-band transitions: intrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

34 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!!

35 Dynamics of electrons in a band Negative effective mass!

36 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.

37 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 )

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

39 Lecture 14: Intrinsic and extrinsic semiconductors recap of the effective mass method analysis of E(k) and it s 1st and nd deriviates band-to-band transisions: intrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

40 Band-to-band transitions 40

41 Band-to-band transitions valence band conduction band

42 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

43 Band-to-band transitions valence band conduction band

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

45 Band-to-band transitions

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

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

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

49 Band-to-band transitions

50 Band-to-band transitions CBM μ VBM

51 Band-to-band transitions In quantum wells

52 Lecture 14: Intrinsic and extrinsic semiconductors recap of the effective mass method analysis of E(k) and it s 1st and nd deriviates band-to-band transiotions: intrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

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

54 Hydrogen atom - Bohr model ) (4 ) (4 energy: Total 4 1 Kineticenergy: 4 4 Potentialenergy: ) ( 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

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

56 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 ev m * n m0 0 Ks ev

57 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

58 Lecture 14: Intrinsic and extrinsic semiconductors recap of the effective mass method analysis of E(k) and it s 1st and nd deriviates band-to-band transiotions: intrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

59 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

60 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.

61 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

62 N- and p-type semiconductors

63 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.

64 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

65 Lecture 14: Intrinsic and extrinsic semiconductors recap of the effective mass method analysis of E(k) and it s 1st and nd deriviates band-to-band transiotions: intrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

66 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:

67 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.

68 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.

69 Equilibrium charge carrier concentration in semiconductors Example Si is doped with 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 cm 17 n 0 n e i ( E F E i ) kt 17 n 10 EF Ei ktln ln eV 10 n i

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

71 Lecture 14: Intrinsic and extrinsic semiconductors recap of the effective mass method analysis of E(k) and it s 1st and nd deriviates band-to-band transiotions: intrinsic semiconductors hydrogen-like impurities n- and p-type semiconductors equilibrium charge carrier concentration carriers in non-eqilibrium conditions: diffusion, generation and recombination

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

73 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.

74 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

75 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)

76 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

77 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

78 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

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

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

81 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

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

83 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.

84 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

85 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