SEMICONDUCTOR PHYSICS

Transcription

1 SEMICONDUCTOR PHYSICS by Dibyendu Chowdhury

2 Semiconductors The materials whose electrical conductivity lies between those of conductors and insulators, are known as semiconductors. Silicon Germanium Cadmium Sulphide 1.1 ev 0.7 ev 2.4 ev Silicon is the most widely used semiconductor. Semiconductors have negative temperature coefficients of resistance, i.e. as temperature increases resistivity deceases

3 Energy Band Diagram Conduction electrons

4 Energy Band Diagram Forbidden energy band is small for semiconductors. Less energy is required for electron to move from valence to conduction band. A vacancy (hole) remains when an electron leaves the valence band. Hole acts as a positive charge carrier.

5 Both silicon and germanium are tetravalent, i.e. each has four electrons (valence electrons) in their outermost shell. Intrinsic Semiconductor A semiconductor, which is in its extremely pure form, is known as an intrinsic semiconductor. Silicon and germanium are the most widely used intrinsic semiconductors. Each atom shares its four valence electrons with its four immediate neighbours, so that each atom is involved in four covalent bonds.

6 Intrinsic Semiconductor When the temperature of an intrinsic semiconductor is increased, beyond room temperature a large number of electron-hole pairs are generated. Since the electron and holes are generated in pairs so, Free electron concentration (n) = concentration of holes (p) = Intrinsic carrier concentration (n i )

7 Extrinsic Semiconductor Pure semiconductors have negligible conductivity at room temperature. To increase the conductivity of intrinsic semiconductor, some impurity is added. The resulting semiconductor is called impure or extrinsic semiconductor. Impurities are added at the rate of ~ one atom per 10 6 to semiconductor atoms. The purpose of adding impurity is to increase either the number of free electrons or holes in a semiconductor.

8 Extrinsic Semiconductor Two types of impurity atoms are added to the semiconductor Atoms containing 5 valance electrons (Pentavalent impurity atoms) e.g. P, As, Sb, Bi Atoms containing 3 valance electrons (Trivalent impurity atoms) e.g. Al, Ga, B, In N-type semiconductor P-type semiconductor

9 N-type Semiconductor The semiconductors which are obtained by introducing pentavalent impurity atoms are known as N-type semiconductors. Examples are P, Sb, As and Bi. These elements have 5 electrons in their valance shell. Out of which 4 electrons will form covalent bonds with the neighbouring atoms and the 5 th electron will be available as a current carrier. The impurity atom is thus known as donor atom. In N-type semiconductor current flows due to the movement of electrons and holes but majority of through electrons. Thus electrons in a N-type semiconductor are known as majority charge carriers while holes as minority charge carriers.

10 P-type Semiconductor The semiconductors which are obtained by introducing trivalent impurity atoms are known as P-type semiconductors. Examples are Ga, In, Al and B. These elements have 3 electrons in their valance shell which will form covalent bonds with the neighbouring atoms. The fourth covalent bond will remain incomplete. A vacancy, which exists in the incomplete covalent bond constitute a hole. The impurity atom is thus known as acceptor atom. In P-type semiconductor current flows due to the movement of electrons and holes but majority of through holes. Thus holes in a P- type semiconductor are known as majority charge carriers while electrons as minority charge carriers.

11 Fermi Energy The Fermi energy is a quantum mechanical concept and it usually refers to the energy of the highest occupied quantum state in a system of fermions at absolute zero temperature.. Fermi Level The Fermi level (E F ) is the maximum energy, which can be occupied by an electron at absolute zero (0 K).

12 Fermi Energy Diagram for Intrinsic Semiconductors Forbidden Energy Gap Fermi Level (E F ) The Fermi level (E F ) lies at the middle of the forbidden energy gap.

13 Fermi Energy Diagram for N-type Semiconductors Fermi Level (E F ) Energy (ev) Donor Level Fermi Level (E F ) The Fermi level (E F ) shifts upwards towards the bottom of the conduction band.

14 Fermi Energy Diagram for P-type Semiconductors Energy (ev) Acceptor Level Fermi Level (E F ) Fermi Level (E F ) The Fermi level (E F ) shifts downwards towards the top of the valance band.

15 Mass Action Law Addition of n-type impurities decreases the number of holes below a level. Similarly, the addition of p-type impurities decreases the number of electrons below a level. It has been experimentally found that Under thermal equilibrium for any semiconductor, the product of no. of holes and the no. of electrons is constant and independent of amount of doping. This relation is known as mass action law n 2. p ni where n = electron concentration, p = hole concentration and n i = intrinsic concentration

16 Charge carrier concentration in N-type and P-type Semiconductors The free electron and hole concentrations are related by the Law of Electrical Neutrality i.e. Total positive charge density is equal to the total negative charge density Let N D = Concentration of donor atoms = no. of positive charges/m 3 contributed by donor ions p = hole concentration N A =Concentration of acceptor atoms n = electron concentration By the law of electrical neutrality N D + p = N A + n

17 For N-Type semiconductor N A = 0 i.e. Concentration of acceptor atoms And n>>p, then N D + 0 = 0 + n N D = n i.e. in N-type, concentration of donor atoms is equal to the concentration of free electrons. According to Mass Action Law n 2. p ni p n / n n / 2 i 2 i N D

18 For P-Type semiconductor N D = 0 i.e. Concentration of donor atoms And p>>n, then N A + 0 = 0 + p N A = p i.e. in P-type, concentration of acceptor atoms is equal to the concentration of holes. According to Mass Action Law n 2. p ni n n / p n / 2 i 2 i N A

19 The Concept of Effective Mass : Comparing Free e - in vacuum In an electric field m o =9.1 x Free electron mass An e - in a crystal In an electric field In a crystal If the same magnitude of electric field is applied to both electrons in vacuum and inside the crystal, the electrons will accelerate at a different rate from each other due to the existence of different potentials inside the crystal. The electron inside the crystal has to try to make its own way. So the electrons inside the crystal will have a different mass than that of the electron in vacuum. m =? m * effective mass This altered mass is called as an effective-mass.

20 What is the expression for m * Particles of electrons and holes behave as a wave under certain conditions. So one has to consider the de Broglie wavelength to link partical behaviour with wave behaviour. Partical such as electrons and waves can be diffracted from the crystal just as X-rays. Certain electron momentum is not allowed by the crystal lattice. This is the origin of the energy band gaps. n 2d sin n = the order of the diffraction λ = the wavelength of the X-ray d = the distance between planes θ = the incident angle of the X-ray beam

21 n = 2d The waves are standing waves = 2 k The momentum is P = k (1) is the propogation constant The energy of the free electron can be related to its momentum E E (2) 2 h P P = = 2m free e - mass, m h h 2m 2m = h 2 E k2 2m k (2 ) The energy of the free e - is related to the k By means of equations (1) and (2) certain e - momenta are not allowed by the crystal. The velocity of the electron at these momentum values is zero. Energy momentum E versus k diagram is a parabola. Energy is continuous with k, i,e, all energy (momentum) values are allowed. E versus k diagram or Energy versus momentum diagrams k

22 To find effective mass, m * We will take the derivative of energy with respect to k ; de dk d E 2 dk 2 k m m m* is determined by the curvature of the E-k curve - m* is inversely proportional to the curvature Change m* instead of m m * d E dk This formula is the effective mass of an electron inside the crystal.

23 Direct an indirect-band gap materials : Direct-band gap s/c s (e.g. GaAs, InP, AlGaAs) CB E For a direct-band gap material, the minimum of the conduction band and maximum of the valance band lies at the same momentum, k, values. e - + k When an electron sitting at the bottom of the CB recombines with a hole sitting at the top of the VB, there will be no change in momentum values. VB Energy is conserved by means of emitting a photon, such transitions are called as radiative transitions.

24 Indirect-band gap s/c s (e.g. Si and Ge) E For an indirect-band gap material; the minimum of the CB and maximum of the VB lie at different k-values. When an e - and hole recombine in an indirect-band gap s/c, phonons must be involved to conserve momentum. CB Phonon + k e - Eg Atoms vibrate about their mean position at a finite temperature.these vibrations produce vibrational waves inside the crystal. Phonons are the quanta of these vibrational waves. Phonons travel with a velocity of sound. VB Their wavelength is determined by the crystal lattice constant. Phonons can only exist inside the crystal.

25 The transition that involves phonons without producing photons are called nonradiative (radiationless) transitions. These transitions are observed in an indirect band gap s/c and result in inefficient photon producing. So in order to have efficient LED s and LASER s, one should choose materials having direct band gaps such as compound s/c s of GaAs, AlGaAs, etc

26 .:: CALCULATION For GaAs, calculate a typical (band gap) photon energy and momentum, and compare this with a typical phonon energy and momentum that might be expected with this material. photon phonon E(photon) = Eg(GaAs) = 1.43 ev E(phonon) = h = hv s / λ E(photon) = h c= 3x10 8 m/sec = hc / λ = hv s / a0 λ (phonon) ~a0 = lattice constant =5.65x10-10 m P = h / λ h=6.63x10-34 J-sec Vs= 5x10 3 m/sec ( velocity of sound) λ (photon)= 1.24 / 1.43 = 0.88 μm P(photon) = h / λ = 7.53 x kg-m/sec E(phonon) = hv s / a 0 =0.037 ev P(phonon)= h / λ = h / a 0 = 1.17x10-24 kg-m/sec

27 Photon energy = 1.43 ev Phonon energy = 37 mev Photon momentum = 7.53 x kg-m/sec Phonon momentum = 1.17 x kg-m/sec Photons carry large energies but negligible amount of momentum. On the other hand, phonons carry very little energy but significant amount of momentum.

28 Direct-band gap s/c s (e.g. GaAs, InP, AlGaAs) CB E Positive and negative effective mass m * d E dk The sign of the effective mass is determined directly from the sign of the curvature of the E-k curve. e - + k The curvature of a graph at a minimum point is a positive quantity and the curvature of a graph at a maximum point is a negative quantity. Particles(electrons) sitting near the minimum have a positive effective mass. VB Particles(holes) sitting near the valence band maximum have a negative effective mass. A negative effective mass implies that a particle will go the wrong way when an extrernal force is applied.

29 Energy (ev) Energy (ev) 4 GaAs Conduction band 4 Si Conduction band ΔE= Eg 1 Eg Valance band [111] 0 [100] k -2 Valance band [111] 0 [100] k Energy band structures of GaAs and Si

30 Energy (ev) 4 3 GaAs Conduction band Band gap is the smallest energy separation between the valence and conduction band edges. 2 ΔE= Eg The smallest energy difference occurs at the same momentum value -1-2 Valance band [111] 0 [100] k Direct band gap semiconductor Energy band structure of GaAs

31 Energy (ev) The smallest energy gap is between the top of the VB at k=0 and one of the CB minima away from k=0 4 3 Si Conduction band 2 Indirect band gap semiconductor Band structure of AlGaAs? Effective masses of CB satellites? Heavy- and light-hole masses in VB? Valance band Eg [111] 0 [100] k Energy band structure of Si

32 E E g direct transition k

33 E E g direct transition k

34 E E g k

35 E E g indirect transition k

36 E E g indirect transition k