10-11 a-energy_bands

1. Pertemuan 10 - 11 FISIKA ZAT PADAT ENERGY BANDS Iwan Sugihartono, M.Si Jurusan Fisika Fakultas Matematika dan Ilmu Pengetahuan Alam

5. REVIEW OF ENERGY BANDS (1) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | In crystalline solids, the atoms are “assembled” in a periodical arrangement, in such a way as to minimize the energy of the system… Kasap, S.O., Principles of electrical engineering materials and devices , McGraw-Hill, 1997 Example: NaCl crystal (ionic bound) In the solid, the separation between the constituting atoms is comparable to the atomic size, so the properties of the individual atoms are altered by the presence of neighbouring atoms.

6. REVIEW OF ENERGY BANDS (2) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | The permitted energies that an electron can occupy in the isolated atoms are split into energy bands as the atoms get closer to each other. This can also be visualized in terms of an overlap of the electron wave functions (Streetman section 3.1.2). Holden A., The nature of solids , Dover Publications, 1965 Outer shell Since the solid is made of a very large number of atoms (N), these bands are actually constituted by N levels of energies (almost a continuum!). The position of upper and lower extreme of each band is however independent of N (depends only on the materials).

7. REVIEW OF ENERGY BANDS (3) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Mathematically, it means solving the time-independent Schrödinger’s equation: where U(r) is the periodic effective potential energy that describes the arrangement of atoms in the crystal. e.g. a 1-D crystal could for instance be represented in the following manner: a + b a + b a b U o x = 0 Adapted from: R. Eisberg, R. Resnick, Quantum physics of atoms, molecules, solids, nuclei, and particles (2 nd ed.) , Wiley, 1985

8. REVIEW OF ENERGY BANDS (4) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | The solution to the equation is usually given in the form of a “band diagram” E vs k . Yu, P.Y., Cardona, M., Fundamentals of semiconductors , Springer, 2005 e.g. GaAs crystal Cohen M.L., Chelikowski, J.R., Electronic Structure and Optical Properties of semiconductors , Springer, 1989 E g ~ 1.4 eV Actual band diagram For device description a much more simple and practical representation is typically used: E g E c E v

9. REVIEW OF ENERGY BANDS (5) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Direct band gap semiconductors (e.g. GaAs, InP, InAs, GaSb) Cohen M.L., Chelikowski, J.R., Electronic Structure and Optical Properties of semiconductors , Springer, 1989 E g ~ 1.4 eV The minimum of the conduction band occurs at the same k value as the valence band maximum.

10. REVIEW OF ENERGY BANDS (6) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Indirect band gap semiconductors (e.g. Si, Ge, AlAs, GaP, AlSb) Cohen M.L., Chelikowski, J.R., Electronic Structure and Optical Properties of semiconductors , Springer, 1989 E g ~ 2.3 eV The minimum of the conduction band does not occur at the same k value as the valence band maximum. An electron promoted to the conduction band requires a change of its momentum to make the transition to the valence band (typ. occurs via lattice vibrations).

13. TERNARY III-V ALLOYS (1) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | The group III lattice sites are occupied by a fraction x of atoms III 1 and a fraction ( 1-x ) of atoms III 2 . Arsenides: e.g. InGaAs (used in the active regions of high-speed electronic devices, IR lasers, and long-wavelength quantum cascade lasers) Phosphides: e.g. InGaP (GaAs-based quantum well devices such as red diode lasers) Antimonides: e.g. AlGaSb (employed in high-speed electronic and infrared optoelectronic devices) Nitrides: e.g. InGaN (key constituent in the active regions of blue diode lasers and LED’s)

14. TERNARY III-V ALLOYS (2) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | The group V lattice sites are occupied by a fraction y of atoms V 1 and a fraction ( 1-y ) of atoms V 2 . Arsenides Antimonides: e.g. InAsSb (smallest band gap of all III-V’s, very important material for mid-infrared optoelectronic devices) Arsenides Phosphides: e.g. GaAsP (often used for red LED’s) Phosphides Antimonides: e.g. GaPSb

15. TERNARY III-V ALLOYS (4) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | http://www.rpi.edu/~schubert/Light-Emitting-Diodes-dot-org/chap07/F07-06-R.jpg Band Gap Engineering: Arsenides, Phosphides, Antimonides Ga x In 1-x As Ga x In 1-x P

16. TERNARY III-V ALLOYS (3) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | http://www.onr.navy.mil/sci_tech/31/312/ncsr/materials/gan.asp Band Gap Engineering: Nitrides 1.24  m 620 nm 413 nm 310 nm 248 nm 207 nm 

18. Quaternary III-V Alloys (2) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Panish, M.B., Temkin, H., Gas source Molecular Beam Epitaxy , Springer, 1993 Wide variety of compositions (hence various emission wavelengths) are lattice matched to either GaAs or InP… The case of Ga x In 1-x As y P 1-y

19. DESIGN CONSIDERATIONS 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Most practical devices consists in heterostructures, i.e. layered thin film structures made of dissimilar materials deposited on top of each other (e.g. quantum well). Based on the device requirements, the designer will select the proper alloy, while keeping in mind that typical substrates consist in binary compound (such as GaAs, InP, InP, GaSb) in order to minimize the effects of lattice mismatch. The misfit is defined as: a s : lattice constant of the substrate material a f : lattice constant of the film material

20. LATTICE-MATCHED STRUCTURE 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | By definition, f = 0 , i.e. both the substrate material and the film material have the same lattice constant. A misfit |f| < 5 × 10 -4 is generally considered very good, and for practical purposes is assumed lattice-matched (e.g. AlGaAs/GaAs structures) Ohring, M., The Materials Science of Thin Films , Academic Press, 1992

21. STRAINED STRUCTURES 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Ohring, M., The Materials Science of Thin Films , Academic Press, 1992 f ≠ 0 (i.e. a s ≠ a f ) For relatively thin film thicknesses, cubic crystals will distort (strain develops within the layer) to achieve the same in-plane lattice constant. Such a layer is referred to as pseudomorphic . f < 0 : compressive strain f > 0 : tensile strain

22. RELAXED STRUCTURE 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Beyond a certain critical thickness h c , it is energetically more favourable for the film to “relax”, i.e. achieve a state where its lattice constant tends towards its unstrained value. In the process, defects known as misfit dislocations form at the interface… Tu, K-N, Mayer, J.W, Feldman, L.C., Electronic thin film science for electrical engineers and materials scientists , McMillan, 1992 Burgers vector: atomic displacement needed to generate the dislocation

23. CRITICAL THICKNESS (1) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | h c is obtained by minimizing E tot with respect to the strain. This results in a transcendent equation: where  is the Poisson’s ratio of the film material.

24. CRITICAL THICKNESS (2) 15/03/11 © 2010 Universitas Negeri Jakarta | www.unj.ac.id | Mayer, J.W, Lau, S.S., Electronic Materials Science: For integrated circuits in Si and GaAs , McMillan, 1990

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