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BAND THEORY OF SOLIDS Page * LUẬN ĐIỂM VẬT LÝ CHẤT RẮN CƠ BẢN Các lọai chất rắn Ion, cộng hóa trị, kim loại, Van der Wall và Hydro Lý thuyết cổ điển về tính dẫn Mật độ dòng j, tốc độ dòng vd, trở lực[.]

LUẬN ĐIỂM VẬT LÝ CHẤT RẮN CƠ BẢN • Các lọai chất rắn – Ion, cộng hóa trị, kim loại, Van der Wall Hydro • Lý thuyết cổ điển tính dẫn – Mật độ dịng j, tốc độ dịng vd, trở lực  • Lý thuyết dải biểu đồ dải lượng – Các mức lượng nguyên tử riêng biệt tạo thành dải lượng xích lại gần tinh thể – Hàm Fermi dải lượng lấp đầy – Các dải lượng kim loại, chất cách điện bán dẫn – Tạp chất cho nhận Donor and Acceptor dopants (Hiệu ứng đường hầm Hall Effect) • Linh kiện Devices – pn junction, diode, LED, solar cell, laser Phys 320 - Baski Page Tính chất chất rắn liên kết ion • Tạo lực hút tĩnh điện Coulomb ion – Ví dụ: cation kiềm nhóm I anion nhóm VII: Na+ Cl- • Năng lượng liên kết lớn (2-4 eV/ atom) – Nhiệt độ nóng chảy nhiệt độ sơi cao • Độ dẫn điện thấp – Khơng có electron tự tạo dịng điện • Trong suốt với ánh sáng nhìn thấy – Năng lượng photon thấp để giải phóng electron • Hịa tan chất lỏng phân cực (nước) – Lưỡng cực nước hút ion Phys 320 - Baski Page Chất rắn liên kết ion Năng lượng đẩy  1/rm Năng lượng tổng Lực hút Coulomb  -1/r • Potential Energy: U = Uh (+,–) + U đẩy (–, –) Phys 320 - Baski Page CẤU TRÚC TINH THỂ Simple Cubic FCC structure: NaCl Body-Centered Cubic Face-Centered Cubic Na+ Cl- Phys 320 - Baski Page CHẤT RẮN CỘNG HÓA TRỊ • Ví dụ: ngun tố nhóm IV (C, Si) nhóm III-V (GaAs, InSb) • Formed by strong, localized bonds with stable, closed-shell structures • Larger cohesive energies than for ionic solids (4-7 eV/atom) – Leads to higher melting and boiling points • Low electrical conductivity – Due to energy band gap that charged carriers must overcome in order to conduct Phys 320 - Baski Page Types of Solids: Example Crystalline Structures Graphite Planar sp2 bonding (good lubricant) Vertical -bonds Phys 320 - Baski Diamond Tetrahedral sp3 bonding (very hard!) Bond angle = 109.5º Page Types of Solids: Metal • Formed by Coulombic attraction between (+) lattice ions and (–) electron “gas.” • Metallic bonds allows electrons to move freely through lattice • Smaller cohesive energy (1-4 eV) • High electrical conductivity • Absorbs visible light (non-transparent, “shiny” due to re-emission) • Good alloy formation (due to non-directional metallic bonds) Phys 320 - Baski Page Classical Theory of Conduction (E&M Review) Macroscopic dq Current: i  (Amps) dt q  idt V i R L R A Microscopic  di Current Density: J   (A/m ) dA    i  J dA  E  J   E where  resistivity    J n e v d  m ne   conductivity where n carrier density vd drift velocity where  scattering time • Drift velocity vd is net motion of electrons (0.1 to 10-7 m/s) • Scattering time  is time between electron-lattice collisions Phys 320 - Baski Page Classical Theory of Conduction: Electron Motion • Electron travels at fast velocities for a time  and then “collides” with the crystal lattice • Results in a net motion opposite to the E field with drift velocity vd • Scatter time  decreases with increasing temperature T, i.e more scattering at higher temperatures (leads to higher resistivity) Phys 320 - Baski Page Classical Theory of Conduction: Resistivity vs Temp • Temperature dependence of resistivity FE ma E m e e      J ne vd ne (a ) n ne  • Metal: Resistance increases with Temperature • Why? Temp  , n same (same # conduction electrons)   • Semiconductor: Resistance decreases with Temperature • Why? Temp  , n (“free-up” carriers to conduct)   Phys 320 - Baski Page 10 Band Theory: “Bound” Electron Approach • For the total number N of atoms in a solid (1023 cm–3), N energy levels split apart within a width E – Leads to a band of energies for each initial atomic energy level (e.g 1s energy band for 1s energy level) Two atoms Electrons must occupy different energies due to Phys 320 -Pauli Baski Exclusion principle Six atoms Solid of N atoms Page 12 Band Diagram: Fermi-Dirac “Filling” Function • Probability of electrons (fermions) to be found at various energy levels f FD  E   • At RT, E – EF = 0.05 eV  f(E) = 0.12 E – EF = 7.5 eV  f(E) = 10 –129  E  EF  e T=0K kT 1 • Exponential dependence has HUGE effect! Moderate T High T • At T = K, electrons have 100% probability to be below Fermi energy EF and 0% probability above EF At T > K, probabilities decrease below EF and increase above EF, causing the step function to “smear out.” : http://jas.eng.buffalo.edu/education/semicon/fermi/functionAndStates/functionAndStates.html PhysFermi 320 - Baski Page 13 Band Diagram: Metal “Fill” the energy band with electrons EC,V Fermi “filling” function Energy band to be “filled” EF T=0K EC,V EF Moderate T • At T = 0, energy levels below EF are filled with electrons, while all levels above EF are empty • Electrons are free to move into “empty” states of conduction band with only a small electric field E, leading to high electrical conductivity! • At T > 0, electrons have a probability to be thermally “excited” from below the Fermi energy to above it Phys 320 - Baski Page 14 Band Diagram: Insulator with large Egap T>0 Conduction band (Empty) EC Egap EF Valence band (Filled) EV • At T = 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity – Fermi energy EF is at midpoint of large energy gap (2-10 eV) between conduction and valence bands • At T > 0, electrons are NOT thermally “excited” from valence to conduction band, leading to zero conductivity Phys 320 - Baski Page 15 Band Diagram: Semiconductor with moderate Egap T>0 Conduction band (Partially Filled) EF Valence band EC EV (Partially Empty) • At T = 0, valence band is filled with electrons and conduction band is empty, leading to zero conductivity • At T > 0, electrons thermally “excited” from valence to conduction band, leading to partially empty valence and partially filled conduction bands What happens to the conductivity for T > 0? How Phys 320would - Baski the band diagram look for lower & higher temperatures? Page 16 Band Diagram: Donor Dopant in Semiconductor • Increase the conductivity of a semiconductor by adding a small amount of another material called a dopant (instead of heating it!) • For group IV Si, add a group V element to “donate” an electron and make n-type Si (more negative electrons!) • “Extra” electrons donated from donor energy level ED just below EC – Resultant electrons in conduction band increase conductivity by increasing free carrier density n EC EF n-type Si ED EV • Fermi level EF moves up because there are more carriers Fermi & Doping: http://jas.eng.buffalo.edu/education/semicon/fermi/bandAndLevel/fermi.html Phys 320 Function - Baski Page 17 Band Diagram: Acceptor Dopant in Semiconductor • For Si, add a group III element to “accept” an electron and make p-type Si (more positive “holes”!) • “Missing” electrons trapped in acceptor energy level EA just above EV – Resultant holes in valence band increase conductivity • Fermi level EF moves down because there are fewer carriers EC EF EA EV p-type Si Will the Fermi level move if the temperature is changed? Phys 320 - Baski Page 18 Semiconductor: Dopant Density via Hall Effect • Why Useful? Determines carrier type (electron vs hole) and carrier density n for a semiconductor • How? Place semiconductor into external B field, push current along one axis, and measure induced Hall voltage VH along perpendicular axis Carrier density n = (current I) (magnetic field B) (carrier charge q) (thickness t)(Hall voltage VH) • Derived from Lorentz equation FE (qE) = FB (qvB) Hole + charge Phys 320 - Baski    FB qv B Electron – charge Page 19 pn Junction: Band Diagram • At equilibrium, Fermi levels (or charge carrier densities) must equalize • Hence, electrons move from n to p-side (diffusion process) • Depletion zone occurs at junction where immobile charged ion cores remain • Results in a built-in electric field (103 to 105 V/cm), which opposes further diffusion pn regions “touch” & free carriers move EC EF n-type electrons EF EV p-type pn regions in equilibrium EC EF EV PIN 320 junction: Phys - Baskihttp://jas.eng.buffalo.edu/education/pin/pin/# –– – +–– – + + + – –– + ++– + ++–– ++ Depletion Zone Page 20