VIEWS: 2 PAGES: 54 CATEGORY: Education POSTED ON: 10/4/2012
Semiconductor Physics Introduction • Semiconductors are materials whose electronic properties are intermediate between those of Metals and Insulators. • They have conductivities in the range of 10 -4 to 10 +4S/m. • The interesting feature about semiconductors is that they are bipolar and current is transported by two charge carriers of opposite sign. • These intermediate properties are determined by 1.Crystal Structure bonding Characteristics. 2.Electronic Energy bands. • Silicon and Germanium are elemental semiconductors and they have four valence electrons which are distributed among the outermost S and p orbital's. • These outer most S and p orbital's of Semiconductors involve in Sp3 hybridanisation. • These Sp3 orbital's form four covalent bonds of equal angular separation leading to a tetrahedral arrangement of atoms in space results tetrahedron shape, resulting crystal structure is known as Diamond cubic crystal structure Semiconductors are mainly two types 1. Intrinsic (Pure) Semiconductors 2. Extrinsic (Impure) Semiconductors Intrinsic Semiconductor • A Semiconductor which does not have any kind of impurities, behaves as an Insulator at 0k and behaves as a Conductor at higher temperature is known as Intrinsic Semiconductor or Pure Semiconductors. • Germanium and Silicon (4th group elements) are the best examples of intrinsic semiconductors and they possess diamond cubic crystalline structure. Intrinsic Semiconductor Valence Cell Covalent bonds Si Si Si Si Si Conduction band KE of Ec Electron = E - Ec Ec E Ef Electron Fermi energy level energy Ev KE of Hole = Valence band Ev - E Distance Carrier Concentration in Intrinsic Semiconductor When a suitable form of Energy is supplied to a Semiconductor then electrons take transition from Valence band to Conduction band. Hence a free electron in Conduction band and simultaneously free hole in Valence band is formed. This phenomenon is known as Electron - Hole pair generation. In Intrinsic Semiconductor the Number of Conduction electrons will be equal to the Number of Vacant sites or holes in the valence band. Calculation of Density of Electrons Let ‘dn’ be the Number of Electrons available between energy interval ‘E and E+ dE’ in the Conduction band dn Z ( E )dE F ( E ) top of the band n z ( E ) F ( E )dE.......... 1) Ec ......( Where Z(E) dE is the Density of states in the energy interval E and E + dE and F(E) is the Probability of Electron occupancy. We know that the density of states i.e., the number of energy states per unit volume within the energy interval E and E + dE is given by 4 3 1 Z ( E )dE 3 (2m) 2 E 2 dE h 4 3 1 Z ( E )dE 3 (2me ) 2 E 2 dE h Since the E starts at the bottom of the Conduction band E c 4 3 1 Z ( E )dE 3 (2me ) 2 ( E Ec ) 2 dE h Probability of an Electron occupying an energy state E is given by 1 F (E) E Ef 1 exp( ) kT For all possible temperatures E EF kT 1 F (E) E Ef exp( ) kT E EF EF E F ( E ) exp ( ) exp( ) kT kT Substitute Z(E) and F(E) values in Equation (1) top of the band n z ( E ) F ( E )dE Ec 4 3 1 E E n 3 (2me ) 2 ( E Ec ) 2 exp( F )dE Ec h kT 4 3 n 3 (2me ) ( E Ec ) 2 exp( E F E )dE 1 2 h Ec kT 4 E 3 1 E n 3 (2me ) 2 exp( F ) ( E Ec ) 2 exp( )dE.....(2) h kT Ec kT To solve equation 2, let us put E Ec x E Ec x dE dx 4 E 3 1 EF 2 n 3 (2me ) exp( ) ( E Ec ) exp( 2 )dE h kT 0 kT 4 3 EF 1 E x n 3 (2me ) 2 exp( ) ( x) 2 exp ( c )dx h kT 0 kT 4 EF Ec 3 1 x 2 n 3 (2me ) exp( ) ( x) exp ( 2 3 )dx.....( ) h kT 0 kT 1 1 x 3 2 we know that ( x) exp( 2 )dE (kT ) 2 0 kT 2 substitute in equation (3) 1 4 E F Ec 3 3 2 2 n 3 (2me ) exp( ) {(kT ) 2 } h kT 2 2me kT 3 E F Ec n2 ( 2 2 ) exp( ) h kT The above equation represents Number of electrons per unit volume of the Material Calculation of density of holes Let ‘dp’ be the Number of holes or Vacancies in the energy interval ‘E and E + dE’ in the valence band dp Z ( E )dE {1 F ( E )} Ev p z ( E ){1 F ( E )}dE.......... 1) bottom of the band ......( Where Z(E) dE is the density of states in the energy interval E and E + dE and 1-F(E) is the probability of existence of a hole. Density of holes in the Valence band is 4 3 1 Z ( E ) dE 3 ( 2mh ) E dE 2 2 h Since Ev is the energy of the top of the valence band 4 3 2 1 Z ( E )dE 3 (2m ) ( Ev E ) dE h 2 h Probability of an Electron occupying an energy state E is given by 1 1 F (E) 1 { } E Ef 1 exp( ) kT E E f 1 1 F ( E ) 1 {1 exp( )} kT neglect higher order terms in above exp ansion f or higher T values E Ef 1 F ( E ) exp( ) kT Substitute Z(E) and 1 - F(E) values in Equation (1) Ev p z( E ){1 F ( E )}dE bottom of the band 4 E EF Ev 3 1 2 p 3 (2m ) ( Ev E ) exp( h 2 )dE h kT 4 EF 3 Ev 1 E p 3 (2m ) exp( 2 h ) ( Ev E ) exp( )dE....(2) 2 h kT kT To solve equation 2, let us put Ev E x E Ev x dE dx 4 EF 3 Ev 1 E 2 p 3 (2m ) exp( h ) ( Ev E ) exp( )dE 2 h kT kT 4 EF Ev x 3 0 1 2 p 3 (2m ) exp( h ) ( x) exp( 2 )( dx) h kT kT 4 Ev E F x 3 1 2 p 3 (2m ) exp( h ) ( x) exp( )dx 2 h kT 0 kT 1 4 Ev E F 3 3 2 2 p 3 (2m ) exp(h )(kT ) 2 h kT 2 2m kT Ev E F 3 p 2( h 2 ) exp( 2 ) h kT The above equation represents Number of holes per unit volume of the Material Intrinsic Carrier Concentration In intrinsic Semiconductors n = p Hence n = p = n i is called intrinsic Carrier Concentration ni2 np ni np 2me kT 3 E F Ec 2mh kT 3 Ev E F ni {2 ( 2 2 ) exp( )}{2( 2 2 ) exp( )} h kT h kT 2kT E v Ec 3 3 4 ni 2( 2 ) (me m ) exp( 2 h ) h 2kT 2kT 2 4 3 3 Eg ni 2( 2 ) (me mh ) exp( ) h 2kT Fermi level in intrinsic Semiconductors In intrinsic semiconductors n p 2me kT 3 E F Ec 2mh kT 3 Ev E F 2( 2 2 ) exp( ) 2( 2 2 ) exp( ) h kT h kT 2me kT 3 E F Ec 2mh kT 3 Ev E F ( 2 2 ) exp( )( 2 2 ) exp( ) h kT h kT 2 EF mh 3 Ev E c exp( ) ( ) exp( 2 ) kT me kT taking logarithms on both sides Conduction band Ec Ec E mh me * * Ef Electron energy Ev Valence band Temperature 2 EF 3 mh Ev Ec log( ) ( ) kT 2 me kT 3kT mh 3 E Ec EF log( ) 2 ( v ) 4 me 2 In intrinsic semiconductor w eknow that me mh Ev Ec EF ( ) 2 Thus the Fermi energy level EF is located in the middle of the forbidden band. Extrinsic Semiconductors • The Extrinsic Semiconductors are those in which impurities of large quantity are present. Usually, the impurities can be either 3rd group elements or 5th group elements. • Based on the impurities present in the Extrinsic Semiconductors, they are classified into two categories. 1. N-type semiconductors 2. P-type semiconductors N - type Semiconductors When any pentavalent element such as Phosphorous, Arsenic or Antimony is added to the intrinsic Semiconductor , four electrons are involved in covalent bonding with four neighboring pure Semiconductor atoms. The fifth electron is weakly bound to the parent atom. And even for lesser thermal energy it is released Leaving the parent atom positively ionized. N-type Semiconductor Free electron Si Si P Si Si Impure atom (Donor) The Intrinsic Semiconductors doped with pentavalent impurities are called N-type Semiconductors. The energy level of fifth electron is called donor level. The donor level is close to the bottom of the conduction band most of the donor level electrons are excited in to the conduction band at room temperature and become the Majority charge carriers. Hence in N-type Semiconductors electrons are Majority carriers and holes are Minority carriers. Conduction band Ec Ec E Ed Donor levels Eg Electron energy Ev Valence band Distance Carrier Concentration in N-type Semiconductor • Consider Nd is the donor Concentration i.e., the number of donor atoms per unit volume of the material and Ed is the donor energy level. • At very low temperatures all donor levels are filled with electrons. • With increase of temperature more and more donor atoms get ionized and the density of electrons in the conduction band increases. Density of electrons in conduction band is given by 2me kT 3 EF Ec n 2( 2 2 ) exp( ) h kT The density of Ionized donors is given by Ed E F N d {1 F ( Ed )} N d exp( ) kT At very low temperatures, the Number of electrons in the conduction band must be equal to the Number of ionized donors. 2me kT 3 EF Ec Ed E F 2( 2 2 ) exp( ) N d exp( ) h kT kT Taking logarithm and rearranging we get E F Ec Ed E F 2me kT 3 ( )( ) log N d log 2( )2 kT kT h2 Nd 2 E F ( Ed Ec ) kT log 2me kT 2 3 2( 2 ) h ( Ed Ec ) kT Nd EF log 2me kT 2 3 2 2 2( 2 ) h at.,0k ( E d Ec ) EF 2 At 0k Fermi level lies exactly at the middle of the donor level and the bottom of the Conduction band Density of electrons in the conduction band 2me k T 3 E Ec n 2( 2 ) 2 exp( F ) h kT ( E Ec ) k T Nd { d log } Ec 2me k T 2 3 2 2 2( ) E F Ec h 2 exp( ) exp{ } kT kT 1 E F Ec ( E Ec ) (Nd ) 2 E exp( ) exp{ d log c} 2me k T 2 2 3 1 kT 2k T kT [ 2( ) ] h2 1 E F Ec ( E Ec ) (Nd ) 2 exp( ) exp{ d log } 2me k T 2 2 3 1 kT 2k T [ 2( 2 ) ] h 1 E F Ec (Nd )2 ( E d Ec ) exp( ) exp 2me k T 2 2 3 1 kT 2k T [ 2( ) ] h2 2me kT 3 E F Ec n 2( 2 2 ) exp( ) h kT 1 2m kT 3 (Nd )2 ( E d Ec ) n 2( e 2 ) { exp } 2me kT 2 2 2 3 1 h 2kT [ 2( 2 ) ] h 1 2me kT 3 ( E d Ec ) n 2( N d ) ( 2 2 4 ) exp h 2kT Thus we find that the density of electrons in the conduction band is proportional to the square root of the donor concentration at moderately low temperatures. Variation of Fermi level with temperature To start with ,with increase of temperature Ef increases slightly. As the temperature is increased more and more donor atoms are ionized. Further increase in temperature results in generation of Electron - hole pairs due to breading of covalent bonds and the material tends to behave in intrinsic manner. The Fermi level gradually moves towards the intrinsic Fermi level Ei. P-type semiconductors • When a trivalent elements such as Al, Ga or Indium have three electrons in their outer most orbits , added to the intrinsic semiconductor all the three electrons of Indium are engaged in covalent bonding with the three neighboring Si atoms. • Indium needs one more electron to complete its bond. this electron maybe supplied by Silicon , there by creating a vacant electron site or hole on the semiconductor atom. • Indium accepts one extra electron, the energy level of this impurity atom is called acceptor level and this acceptor level lies just above the valence band. • These type of trivalent impurities are called acceptor impurities and the semiconductors doped the acceptor impurities are called P-type semiconductors. Hole Co-Valent bonds Si Si In Si Si Impure atom (acceptor) Conduction band Ec Ec E Electron Eg energy Acceptor levels Ea Ev Valence band temperature • Even at relatively low temperatures, these acceptor atoms get ionized taking electrons from valence band and thus giving rise to holes in valence band for conduction. • Due to ionization of acceptor atoms only holes and no electrons are created. • Thus holes are more in number than electrons and hence holes are majority carriers and electros are minority carriers in P-type semiconductors. • Equation of continuity: • As we have seen already, when a bar of n-type germanium is illuminated on its one face, excess charge carriers are generated at the exposed surface. • These charge carriers diffuse through out the material. Hence the carrier concentration in the body of the sample is a function of both time and distance. • Let us now derive the differential equation which governs this fundamental relationship. • Let us consider the infinitesimal volume element of area A and length dx as shown in figure. • If tp is the mean lifetime of the holes, the holes lost per sec per unit volume by recombination is p/tp . • The rate of loss of charge within the volume under consideration p eAdx tp eAdxg If g is the thermal rte of generation of hole-electron pairs per unit volume, rate of increase of charge wthin the volume under consideration • If i is the current entering the volume at x and i + di the current leaving the volume at x + dx, then decrease of charge per second from the volume under consideration = di dp eAdx • Because of the above dt p stated three effects the hole density changes with time. • Increase in the number of charges per second within the volume Increase = generation - loss dp p eAdx eAdxg eAdx dI dt p tp Since the hole current is the sum of the diffusion current and the drift current dp I AeDp Ape h E dx Where E is the electric field intensity within the volume. when no external field is applied, under thermal equilibrium condition, the hole density attains a constant value p0. dp under these conditions di 0 and 0 dt p0 g tp this equation indicates that the rate of generation of holes is equal to therate of loss due to recombination under equilibriu m conditions. combain,.eq s ...3,4 & 5. dp ( p p0 ) 2 p d ( pE ) Dp 2 h dt tp x dx This is called equation of conservation of charge or the continuity equation. if p is a function of both t and x, partial derivatives should be used. p ( p p0 ) 2 p ( pE ) Dp h t tp x 2 x if w e are considering holes in the n - type material pn ( p n p0 n ) 2 pn ( pn E ) Dp h t tp x 2 x if w e are considering electrons in the p - type material n p (n p n0 p ) 2n p (n p E ) Dn e t te x 2 x Direct band gap and indirect band gap semiconductors: • We known that the energy spectrum of an electron moving in the presence of periodic potential field is divided into allowed and forbidden zones. • In crystals the inter atomic distances and the internal potential energy distribution vary with direction of the crystal. Hence the E-k relationship and hence energy band formation depends on the orientation of the electron wave vector to the crystallographic axes. • In few crystals like gallium arsenide, the maximum of the valence band occurs at the same value of k as the minimum of the conduction band as shown in below. this is called direct band gap semiconductor. E Conduction E Conduction band band Eg Eg k k Valence band Valence band • In few semiconductors like silicon the maximum of the valence band does not always occur at the same k value as the minimum of the conduction band as shown in figure. This we call indirect band gap semiconductor. • In direct band gap semiconductors the direction of motion of an electron during a transition across the energy gap remains unchanged. • Hence the efficiency of transition of charge carriers across the band gap is more in direct band gap than in indirect band gap semiconductors. Hall effect When a magnetic field is applied perpendicular to a current carrying conductor or semiconductor, voltage is developed across the specimen in a direction perpendicular to both the current and the magnetic field. This phenomenon is called the Hall effect and voltage so developed is called the Hall voltage. Let us consider, a thin rectangular slab carrying current (i) in the x- direction. If we place it in a magnetic field B which is in the y-direction. Potential difference Vpq will develop between the faces p and q which are perpendicular to the z-direction. Z + VH + + + P + + - Y + + + B + + + + + +++ + + Q + +++ + + X P – type semiconductor i Z - _ _ _ _ P_ _ VH + Y _ _ _ _ Q_ _ _ _ _ __ B _ X N – type semiconductor i Magnetic deflecting force F q ( vd B ) Hall eclectic deflecting force F qEH When an equilibrium is reached, the magnetic deflecting force on the charge carriers are balanced by the electric forces due to electric Field. q(vd B) qEH E H ( vd B ) Where vd is drift velocity The relation between current density and drift velocity is J vd ne Where n is the number of charge carriers per unit volume. E H ( vd B ) J EH ( B) ne 1 E H ( JB ) ne E H RH JB 1 EH RH ( Hall,.coef f icien ) t ne JB If VH be the Hall voltage in equilibrium ,the Hall electric field. VH EH d Where d is the width of the slab. EH RH JB 1 VH RH JB d If t is the thicknessof the sample, Then its crosssection is dt and current density I J dt VH RH JBd I VH RH ( ) B t V t RH H IB • Since all the three quantities EH , J and B are measurable, the Hall coefficient RH and hence the carrier density can be found out. • Generally for N-type material since the Hall field is developed in negative direction compared to the field developed for a P- type material, negative sign is used while denoting hall coefficient RH.