CHAPTER 2: ENERGY BANDS & CARRIER CONCENTRATION IN THERMAL EQUILIBRIUM
OUTLINE 2.1 INTRODUCTION: 2.1.1 Semiconductor Materials 2.1.2 Basic Crystal Structure 2.1.3 Basic Crystal Growth technique 2.1.4 Valence Bonds 2.1.5 Energy Bands 2.2 Intrinsic Carrier Concentration 2.3 Donors & Acceptors
2.1.1 SEMICONDUCTOR MATERIALS To understand the characteristic of semiconductor materials you need PHYSICS. Basic Solid-State Physics materials, may be grouped into 3 main classes: (i) Insulators, (ii) Semiconductors, and (iii) Conductors - Refer to your basic electronic devices knowledge (EMT 111/4). - Electrical conductivity σ: ρ=1/σ.
Figure 2.1. Typical range of conductivities for insulators, semiconductors, and conductors.
SEMICONDUCTOR S S ELEMENTS The study of semiconductor materials 19 th century. Early 1950s Ge was the major semiconductor material, and later in early 1960s, Si has become a practical substitute with several advantages: (i) better properties at room temperature, (ii) can be grown thermally high quality silicon oxide, (iii) Lower cost, and (iv) Easy to get, silica & silicates comprises 25% of the Earth s crust.
COMPOUND SEMICONDUCTORS Please refer to the Table 2 (in Sze, pg. 20) Types of compounds: (i) binary compounds - combination of two elements. - i.e GaAs is a III IV. (ii) ternary and quaternary compounds - for special applications purposes. - ternary compounds, i.e alloy semiconductor Al x Ga 1-x As (III IV). - quaternary compounds with the form of A x B 1-x C y D 1-y, so-called combination of many binary & ternary compounds. - more complex processes. GaAs high speed electronic & photonic applications
2.1.2 BASIC CRYSTAL STRUCTURE Lattice the periodic arrangement of atoms in a crystal. Unit Cell represent the entire lattice (by repeating unit cell throughout the crystal) The Unit Cell 3-D unit cell shown in Fig. 2.2. Relationship between this cell & the lattice three vectors: a, b, and c (not be perpendicular to each other, not be equal in length). Equivalent lattice point in 3-D: m, n, and p - integers. R = Fig. 2.2. A generalized primitive unit cell. (1) r r r ma + nb + pc
UNIT CELL Figure 2.3. Three cubic-crystal crystal unit cells. (a)( ) Simple cubic. (b)) Body-centered cubic. (c)( ) Face-centered centered cubic. simple cubic (sc) atom at each corner of the cubic lattice, each atoms has 6 equidistant nearest-neighbor atoms. body-centered cubic (bcc) 8 corner atoms, an atom is located at center of the cube. - each atom has 8 nearest-neighbor atoms. face-centered cubic (fcc) 1 atom at each of the 6 cubic faces in addition to the 8 corner atoms. 12 nearest-neighbor atoms. Large number of elements
THE DIAMOND STRUCTURE Si and Ge have a diamond lattice structure shown in Fig. 2.4. Fig. 2.4(a) a corner atom has 1 nearest neighbor in the body diagonal direction, no neighbor in the reverse direction. Most of III-IV compound semiconductor (e.g GaAs) have Fig. 2.4 (b) structure. a/2 a/2 Fig. 2.4.. (a)( ) Diamond lattice. (b)( Zincblende lattice.
CRYSTAL PLANES & MILLER INDICES Figure 2.6. Miller indices of some important planes in a cubic crystal. Crystal properties along different planes are different electrical & other devices characteristics can be dependent on the crystal orientation use Miller indices. Miller indices are obtained using the following steps: (i) find the interfaces of the plane on the 3 Cartesian coordinates in term of the lattice constant. (ii) Take the reciprocal of these numbers and reduce them to smallest 3 integers having the same ratio. (iii) Enclose the result in parentheses (khl) as the Miller indices for a single plane.
BASIC CRYSTAL GROWTH TECHNIQUE Prof. J. Czochralski, 1885-1953 Refer to EMT 261. 95% electronic industry used Si. Steps from SiO 2 or quartzite. Most common method called Czochralski technique (CZ). Melting point of Si = 1412ºC. Choose the suitable orientation <111> for seed crystal. Si ingot Figure 2.8. Simplified schematic drawing of the Czochralski puller. Clockwise (CW), counterclockwise (CCW).
VALENCE BONDS Recall your basic electronic devices EMT 111/4. Fig. 2.11(a) each atom has 4 ē in the outer orbit, and share these valence ē with 4 neighbors. Sharing of ē called covalent bonding occurred between atoms of same and different elements respectively. Figure 2.11. (a)) A tetrahedron bond. (b)( Schematic two-dimensional representation of a tetrahedron bond. Example: GaAs small ionic contribution that is an electrostatic interactive forces between each Ga + ions and its 4 neighboring As - ions -means that the paired bonding ē spend more time in the As atom than in the Ga atom. Figure 2.12. Figure 2.12. The basic bond representation of intrinsic silicon. (a)( ) A broken bond at Position A, resulting in a conduction electron and a hole. (b)( ) A broken bond at position B.
ENERGY BANDS Energy levels for an isolated hydrogen atom are given by the Bohr model*: 4 m q = 2 2 8ε h n E 0 13. 6 = H 2 n 2 0 Neils Bohr, 1885-1962 Nobel Prize in physics 1922 m o free electron mass (0.91094 x 10-30 kg) 1 ev = 1.6 x 10-19 J q electronic charges: 1.6 x 10-19 C ε 0 free space permittivity (8.85418 x 10-12 F/m) h Planck constant (6.62607 x 10-34 J.s) n positive integer called principle quantum number ev (2) For 1 st energy state or ground state energy level, n = 1, E H = -13.6eV. For the 1 st excited energy level, n = 2. * Find Fundamentals Physics books @ KUKUM Library!! For higher principle quantum number (n 2), energy levels are split.
ENERGY BANDS (cont.) For two identical atoms: When they far apart have same energy. When they are bought closer: split into two energy levels by interaction between the atoms. - as N isolated atoms to form a solid. - the orbit of each outer electrons of different atoms overlap & interact with each other. - this interactions cause a shift in the energy levels (case of two interacting atoms). - when N>>>, an essentially continuous band of energy. This band of N level may extend over a few ev depending on the inter-atomic spacing of the crystal.
ENERGY BANDS (cont.) Equilibrium inter-atomic distance of the crystal. Figure 2.13. The splitting of a degenerate state into a band of allowed energies. Figure 2.14. Schematic presentation of an isolated silicon atom.
ENERGY BANDS (cont.) Fig. 2.15: schematic diagram of the formation of a Si crystal from N isolated Si atoms. Inter-atomic distance decreases, 3s and 3p sub shell of N Si atoms will interact and overlap. At equilibrium state, the bands will split again: 4 quantum states/atom (valence band) 4 quantum states/atom (cond. band) Figure 2.15. Formation of energy bands as a diamond lattice crystal is formed by bringing isolated silicon atoms together.
ENERGY BANDS (cont.) At T = 0K, electrons occupy the lowest energy states: Thus, all states in the valence band (lower band) will be full, and all states in the cond. band (upper band) will be empty. The bottom of cond. band is called E C, and the top of valence band called E V. Bandgap energy E g = (E C E V ). Physically, E g defined as the energy required to break a bond in the semiconductor to free an electron to the cond. band and leave the hole in the valence band. (Please remember this important definition). This is one of a hot-issue in scientific research in the world!!! Bandgap is one of the factors that affect the devices performance.
Energy & Momentum Please recall back your memory in the basic of fundamental PHYSICS!! What is the definition of an Energy in PHYSICS??? What is the definition of Momentum in PHYSICS???
The Energy-Momentum Diagram The energy of free-electron is given by E = p 2 m p momentum, m 0 free-electron mass 2 0 (3) effective mass = m n m n d 2 E dp 2 1 (4) for the narrower parabola (correspond to the larger 2 nd derivative) smaller m n Figure 2.16. The parabolic energy (E)( ) vs. momentum (p)( curve for a free electron. for holes, m n = m p
The Energy-Momentum Diagram Fig. at the RHS shows the simplified energymomentum of a special semiconductor with: - electron effective mass m n = 0.25m 0 in cond. band. - hole effective mass m p = m 0 in the valence band. electron energy is measured upward hole energy is measured downward the spacing between p = 0 and minimum of upper parabola is called bandgap E g. For the actual case, i.e Si and GaAs more complex. Figure 2.17. A schematic energy-momentum diagram for a special semiconductor with m n = 0.25m 0 and m p = m 0.
Si GaAs Fig. 2.18 is similar to Fig. 2.17. For Si, max in the valence band occurs at p = 0, but min of cond. band occurs at p = p c (along [100] direction). In Si, when electron makes transition from max point (valence band) to min point (cond. band) it required: Energy change ( E g ) + momentum change ( p c ). In GaAs: without a change in momentum. Si indirect semiconductor. GaAs direct semiconductor. bandgap Figure 2.18. Energy band structures of Si and GaAs.. Circles (º)( ) indicate holes in the valence bands and dots ( )( ) indicate electrons in the conduction bands.
GaAs - very narrow conduction-band parabola, using (8), we may expected that GaAs smaller effective mass (m n = 0.063m 0 ). Si wider conduction band parabola, using (8), effective mass m n = 0.19m 0. The different between direct and indirect band structures is very important for applications in LED and semiconductor laser. This devices require the types of band structure for efficient generation of photons (you may learn this applications in Chapter 9 PHOTONIC DEVICES).
CONDUCTION Metals/Conductors Semiconductors Insulators (Recall you basic knowledge in EMT 111) Can you imagine their behaviors???
CONDUCTION Metals/Conductors Very low resistivity. Cond. band either is partially filled (i.e Cu) or overlaps in valence band (i.e Zn, Pb). No bandgap. Electron are free to move with only a small applied field. Current conduction can readily occur in conductors. Insulators Valence electrons form strong bonds between neighboring atoms (i.e SiO 2 ). No free electrons to participate in current conduction near room temperature. Large bandgap. Electrical conductivity very small very high resistivity. Semiconductors Much smaller bandgap ~ 1eV. At T=0K, no electrons in cond. band. Poor conductors at low temperatures. E g = 1.12eV (Si) and E g =1.42eV (GaAs) at room temp. & normal atmosphere.
(b) (b) (a) Figure 2.19. Schematic energy band representations of (a)( ) a conductor with two possibilities (either the partially filled conduction band shown at the upper portion or the overlapping bands shown at the lower portion), (b)( ) a semiconductor,, and (c)( ) an insulator. (c)
2.2 INTRINSIC CARRIER CONCENTRATION In thermal equilibrium: - at steady-state condition (at given temp. without any external energy, i.e light, pressure or electric field). Intrinsic semiconductor contains relativity small amounts of impurities compared with thermally generated electrons and holes. Electron density, n number electrons per unit volume. To obtained this in intrinsic s/c evaluated the electron density in an incremental energy range de. Thus, n = integrating density of state, N(E), energy range, F(E), and de from bottom of the cond. band, E C = E = 0 to the top of the cond. band E top.
We may write n as n = E top n( E) de = E top 0 0 (5) N( E) F( E) de Fermi-Dirac distribution function F( E) 1 = 1+ exp( E E F ) / (6) kt n is in cm -3, and N(E) is in (cm 3.eV) -1 k Boltzmann constant ~ 1.38066 x 10-23 J/K, and T in Kelvin. Figure 2.20. Fermi distribution function F(E) ) versus (E( E F ) for various temperatures. E F energy of Fermi level (is the energy at which the probability of occupation by electron is exactly one-half)
Enrico Fermi (1901 1954) Nobel Prize in Physics 1938 There are two possible outcomes: If the result confirms the hypothesis, then you've made a measurement. If the result is contrary to the hypothesis, then you've made a discovery Paul Adrien Maurice Dirac (1902 1984) Nobel Prize in Physics 1933 Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field
For energy, 3kT above or below Fermi energy, then (7) F ( E) e ( E E F ) / kt for ( E E F ) > 3kT (8) F ( E) 1 e ( E E F ) / kt for ( E E F ) < 3kT > 3kT the exponential term larger than 20, and < 3kT smaller than 0.05. At (E E F ) < 3kT the probability that a hole occupies a state located at energy E.
Figure 2.21. Intrinsic semiconductor. (a)( ) Schematic band diagram. (b)( ) Density of states. (c)) Fermi distribution function. (d)( ) Carrier concentration. N(E) = (E) 1/2 for a given electron effective mass. E F located at the middle of bandgap. Upper-shaded in (d) corresponds to the electron density.
From Appendix H in Sze, pg. 540, density of state, N(E) is defined as 2mn N( E) = 4π 2 h substituting (9) and (8) into (5), thus 3/ 2 h Planck constant ~ 6.62607 x 10-34 J.s E (9) X N n C = = N [( E E ) / kt ], and p = N exp [( E E ) kt ] C exp F C V V F / X 2πm h n 2 kt = 12 for Si, and X = 2 3 / 2 N for GaAs V = 3 / 2 2 2πm p kt 2 h (10) N C and N V effective density of state in cond. band & valence band respectively. At room temperature, T = 300K; Effetive density N C N V Si 2.86 x 10 19 cm -3 2.66 x 10 19 cm -3 GaAs 4.7 x 1017 cm-3 7.0 x 10 18 cm -3
Intrinsic carrier density is obtained by: np = n 2 i (mass action law) n 2 i = N C N V exp( E g / kt ) (11) n i = N C N V exp( E g / 2kT ) Where E g = E C -E V Figure 2.22. Intrinsic carrier densities in Si and GaAs as a function of the reciprocal of temperature.
2.3 DONORS & ACCEPTORS Recall your basic knowledge in EMT 111 Electronic Devices. When semiconductor is doped with impurities become extrinsic and impurity energy levels are introduced. Donor : n-type Acceptor : p-type Ionization energy: E D = ε 0 ε S 2 m m n 0 E ε 0 permittivity in vacuum ~ 8.85418 x 10-14 F/cm ε S semiconductor permittivity, and E H ~ Bohr s energy model H (12)
Figure 2.23. Schematic bond pictures for (a)( n-type Si with donor (arsenic) and (b)( p- type Si with acceptor (boron).
Figure 2.24. Measured ionization energies (in ev) ) for various impurities in Si and GaAs. The levels below the gap center are measured from the top of the valence band and are acceptor levels unless indicated by D for donor level. The levels above the gap center are measured from the bottom of the conduction band and are donor levels unless indicated by A for acceptor level.
NON-DEGENERATE SEMICONDUCTOR In previous section, we assumed that Fermi level E F is at least 3kT above E V and 3kT below E C. Such semiconductor called non-degenerate s/c. Complete ionization cond. at shallow donors in Si and GaAs where they have enough thermal energy to supply E D to ionize all donor impurities at room temperature, thus provide the same number of electrons in the cond. band. Under complete ionization cond., electron density may be written as n = N D (13) And for shallow acceptors; p = N A (14) Where N D and N A donor and acceptor concentration respectively. From electron and hole density (10) and (13) & (14), thus (15) E E C F E E F V = kt ln( N = kt ln( N * Higher donor/acceptor concentration smaller E. C V / / N N D A ) )
Non-degenerate Semiconductor Figure 2.25. Schematic energy band representation of extrinsic semiconductors with (a)( donor ions and (b)( acceptor ions.
NON-DEGENERATE SEMICONDUCTOR Much closer to cond. band Figure 2.26. n-type semiconductor.. (a)( ) Schematic band diagram. (b)( ) Density of states. (c)) Fermi distribution function (d)( ) Carrier concentration. Note that np = n i2. *** p-type semiconductor???
NON-DEGENERATE SEMICONDUCTOR To express electron & hole densities in term of intrinsic part (concentration and Fermi level) used as a reference level when discussing extrinsic s/c, thus; n p = = n i n i exp ( exp ( [ EF Ei ) / kt ] [ E E ) / kt ] i F (16) In extrinsic s/c, Fermi level moves towards either bottom of cond. band (n-type) or top of valence band (p-type). It depends on the domination of types carriers. Product of the two types of carriers will remains constant at a given temp.
The impurity that is present is in greater concentration, thus it may determines the type of conductivity in the s/c. Under complete ionization: [ ] [ ] 2 2 2 4 ) ( with / 2 1 / 2 1 i A D p i p D A p n i n A D D A n N N B p n n B N N p n n p B N N n N p N n + = = + = = + = + = + (17) Solve (17) with mass action law, thus (18) (19) Subscript of n and p refer to n and p-type. NON NON-DEGENERATE SEMICONDUCTOR DEGENERATE SEMICONDUCTOR
EXAMPLE A Si ingot is doped with 10 16 arsenic atom/cm 3. (a) Find carrier concentration and the Fermi level at room temperature (T = 300K). (b) E g = 1.12eV (Si). Sketch the level of energy.
Real fab. > 10 15 Figure 2.28. Fermi level for Si and GaAs as a function of temperature and impurity concentration. The dependence of the bandgap on temperature is shown.
Figure 2.29. Electron density as a function of temperature for a Si sample with a donor concentration of 10 15 cm - 3. 15 cm
DEGENERATE SEMICONDUCTOR For a very heavy doped n-type and p-type s/c, E F will be above E C or below E V this refereed to degenerate semiconductor. Approximation of (7) and (8) are no longer use. Electron density (5) may solved numerically. Important aspect of high doping ~ bandgap narrowing effect (reduced the bandgap), and it given by (at T=300K); (20) E g N = 22 18 10 mev
CONCLUSION REMARKS The properties of s/c are determined to a large extend by the crystal structure. Miller indices to describe the crystal surfaces & crystal orientation. The bonding of atoms & electron energy-momentum relationship connection to the electrical properties of semiconductor. Energy band diagram is very important to understand using physics approach why some materials are good and some are poor in term of conductor of electric current. Some external/ internal changing of s/c (temperature and impurities) may drastically vary the conductivity of s/c. The understanding Physics behind the semiconductor behaviours is very important for Microelectronic Engineer to handle the problems as well as to produced a high-speed devices performance.
Motivation I was born not knowing and have had only a little time to change that here and there Richard P. Feynman (1918-1988) Nobel Prize in Physics 1965
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