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SEMICONDUCTOR PHYSICS

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JNTU SNIST PPT ENGINEERING PHYSICS 2
SEMICONDUCTOR PHYSICS

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SEMICONDUCTOR PHYSICS

  1. 1. Semiconducto r Physics
  2. 2. 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 +4 S/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.
  3. 3. • 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
  4. 4. Semiconductors are mainly two types 1. Intrinsic (Pure) Semiconductors 2. Extrinsic (Impure) Semiconductors
  5. 5. 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.
  6. 6. Si Si SiSiSi Valence Cell Covalent bonds Intrinsic Semiconductor
  7. 7. E Ef Ev Valence band Ec Conduction band Ec Electron energy Distance KE of Electron = E - Ec KE of Hole = Ev - E Fermi energy level
  8. 8. 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
  9. 9. )1......(..........)()( )()( bandtheoftop ∫= = cE dEEFEzn EFdEEZdn Calculation of Density of Electrons Let ‘dn’ be the Number of Electrons available between energy interval ‘E and E+ dE’ in the Conduction band 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.
  10. 10. dEEEm h dEEZ ce 2 1 2 3 3 )()2( 4 )( −= ∗π Since the E starts at the bottom of the Conduction band Ec dEEm h dEEZ e 2 1 2 3 3 )2( 4 )( ∗ = π 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 dEEm h dEEZ 2 1 2 3 3 )2( 4 )( π =
  11. 11. )exp()(exp)( )exp( 1 )( restemperatupossibleallFor )exp(1 1 )( kT EE kT EE EF kT EE EF kTEE kT EE EF FF f F f − = − −= − = >>− − + = Probability of an Electron occupying an energy state E is given by
  12. 12. )2.....()exp()()exp()2( 4 )exp()()2( 4 )exp()()2( 4 )()( 2 1 2 3 3 2 1 2 3 3 2 1 2 3 3 bandtheoftop ∫ ∫ ∫ ∫ ∞ ∗ ∞ ∗ ∞ ∗ − −= − −= − −= = c c c c E c F e E F ce E F ce E dE kT E EE kT E m h n dE kT EE EEm h n dE kT EE EEm h n dEEFEzn π π π Substitute Z(E) and F(E) values in Equation (1)
  13. 13. )3.....()(exp)()exp()2( 4 )(exp)()exp()2( 4 )exp()()exp()2( 4 0 2 1 2 3 3 0 2 1 2 3 3 0 2 1 2 3 3 ∫ ∫ ∫ ∞ ∗ ∞ ∗ ∞ ∗ − − = + −= − −= = += =− dx kT x x kT EE m h n dx kT xE x kT E m h n dE kT E EE kT E m h n dxdE xEE xEE cF e cF e c F e c c π π π To solve equation 2, let us put
  14. 14. )exp() 2 (2 } 2 ){()exp()2( 4 2 3 2 2 1 2 3 2 3 3 kT EE h kTm n kT kT EE m h n cFe cF e − = − = ∗ ∗ π ππ )3( 2 )()exp()( 2 1 2 3 0 2 1 equationinsubstitute kTdE kT x xthatknowwe π = − ∫ ∞ The above equation represents Number of electrons per unit volume of the Material
  15. 15. Calculation of density of holes )1......(..........)}(1){( )}(1{)( bandtheofbottom ∫ −= −= Ev dEEFEzp EFdEEZdp Let ‘dp’ be the Number of holes or Vacancies in the energy interval ‘E and E + dE’ in the valence 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.
  16. 16. dEEm h dEEZ h 2 1 2 3 3 )2( 4 )( ∗ = π Density of holes in the Valence band is Since Ev is the energy of the top of the valence band dEEEm h dEEZ vh 2 1 2 3 3 )()2( 4 )( −= ∗π
  17. 17. )exp()(1 exp )}exp(1{1)(1 } )exp(1 1 {1)(1 1 kT EE EF valuesThigherfor ansionaboveintermsorderhigherneglect kT EE EF kT EE EF f f f − =− − +−=− − + −=− − Probability of an Electron occupying an energy state E is given by
  18. 18. )2....()exp()()exp()2( 4 )exp()()2( 4 )}(1){( 2 1 2 3 3 2 1 2 3 3 bandtheofbottom ∫ ∫ ∫ ∞− ∗ ∞− ∗ − − = − −= −= v v E v F h E F vh Ev dE kT E EE kT E m h p dE kT EE EEm h p dEEFEzp π π Substitute Z(E) and 1 - F(E) values in Equation (1)
  19. 19. ∫ ∫ ∫ ∞ ∗ ∞ ∗ ∞− ∗ −− = − −− = − − = −= −= =− 0 2 1 2 3 3 0 2 1 2 3 3 2 1 2 3 3 )exp()()exp()2( 4 ))(exp()()exp()2( 4 )exp()()exp()2( 4 dx kT x x kT EE m h p dx kT xE x kT E m h p dE kT E EE kT E m h p dxdE xEE xEE Fv h vF h E v F h v v v π π π To solve equation 2, let us put
  20. 20. )exp() 2 (2 2 ))(exp()2( 4 2 3 2 2 1 2 3 2 3 3 kT EE h kTm p kT kT EE m h p Fvh Fv h − = − = ∗ ∗ π ππ The above equation represents Number of holes per unit volume of the Material
  21. 21. Intrinsic Carrier Concentration In intrinsic Semiconductors n = p Hence n = p = n i is called intrinsic Carrier Concentration ) 2 exp()() 2 (2 ) 2 exp()() 2 (2 )}exp() 2 (2)}{exp() 2 (2{ 4 3 2 3 2 4 3 2 3 2 2 3 2 2 3 2 2 kT E mm h kT n kT EE mm h kT n kT EE h kTm kT EE h kTm n npn npn g hei cv hei FvhcFe i i i − = − = −− = = = ∗∗ ∗∗ ∗∗ π π ππ
  22. 22. Fermi level in intrinsic Semiconductors sidesbothonlogarithmstaking )exp()() 2 exp( )exp() 2 ()exp() 2 ( )exp() 2 (2)exp() 2 (2 pntorssemiconducintrinsicIn 2 3 2 3 2 2 3 2 2 3 2 2 3 2 kT EE m m kT E kT EE h kTm kT EE h kTm kT EE h kTm kT EE h kTm cv e hF FvhcFe FvhcFe + = − = − − = − = ∗ ∗ ∗∗ ∗∗ ππ ππ
  23. 23. E Ef Ev Valence band Ec Conduction band Ec Electron energy Temperature ** eh mm =
  24. 24. Thus the Fermi energy level EF is located in the middle of the forbidden band. ) 2 ( thatknowtor wesemiconducintrinsicIn ) 2 ()log( 4 3 )()log( 2 32 cv F he cv e h F cv e hF EE E mm EE m mkT E kT EE m m kT E + = = + += + += ∗∗ ∗ ∗ ∗ ∗
  25. 25. 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
  26. 26. 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 Semiconductors
  27. 27. N-type Semiconductor Si Si SiPSi Free electron Impure atom (Donor)
  28. 28. 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
  29. 29. E Ed Ev Valence band Ec Conduction band Ec Electron energy Distance Donor levels Eg
  30. 30. 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.
  31. 31. )exp() 2 (2 2 3 2 kT EE h kTm n cFe − = ∗ π The density of Ionized donors is given by )exp( )}(1{)( kT EE N EFdEEZ Fd d d − = −= At very low temperatures, the Number of electrons in the conduction band must be equal to the Number of ionized donors. )exp()exp() 2 (2 2 3 2 kT EE N kT EE h kTm Fd d cFe − = −∗ π Density of electrons in Conduction band is given by
  32. 32. Taking logarithm and rearranging we get 2 )( 0., ) 2 (2 log 22 )( ) 2 (2 log)(2 ) 2 (2loglog)()( 2 3 2 2 3 2 2 3 2 cd F e dcd F e d cdF e d FdcF EE E kat h kTm NkTEE E h kTm N kTEEE h kTm N kT EE kT EE + = + + = =+− −= − − − ∗ ∗ ∗ π π π k Fermi level lies exactly at the middle of the donor le the bottom of the Conduction band
  33. 33. Density of electrons in the Conduction band kT EE h kTm N kT EE h kTm N kT EE kT EE kT E h kTm N kT EE kT EE kT E h kTm NkTEE kT EE kT EE h kTm n cd e dcF e dcdcF c e dcdcF c e dcd cF cFe 2 )( exp ]) 2 (2[ )( )exp( } ]) 2 (2[ )( log 2 )( exp{)exp( } ]) 2 (2[ )( log 2 )( exp{)exp( } } ) 2 (2 log 22 )( { exp{)exp( )exp() 2 (2 2 1 2 1 2 1 2 3 2 2 1 2 3 2 2 1 2 3 2 2 1 2 3 2 2 3 2 − = − + − = − −+ + = − −+ + = − − = ∗ ∗ ∗ ∗ ∗ π π π π π
  34. 34. kT EE h kTm Nn kT EE h kTm N h kTm n kT EE h kTm n cde d cd e de cFe 2 )( exp) 2 ()2( } 2 )( exp ]) 2 (2[ )( {) 2 (2 )exp() 2 (2 4 3 2 2 1 2 3 2 2 1 2 3 2 2 3 2 2 1 − = − = − = ∗ ∗ ∗ ∗ π π π π 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.
  35. 35. 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
  36. 36. P - type Semiconductors • When a trivalent elements such as Al, Ga or Indium have three electrons, added to the Intrinsic Semiconductor all the three electrons of these are involved in Covalent bonding with the three neighboring Si atoms. • These like compound 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 with the acceptor impurities are called P-type Semiconductors.
  37. 37. Si Si SiInSi Hole Co-Valent bonds Impure atom (acceptor)
  38. 38. E Ea Ev Valence band Ec Conduction band Ec Electron energy distance Acceptor levels Eg
  39. 39. • 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
  40. 40. Then current density Then conductivity )1.........( E nev E J EJ d = = = σ σ σ )2........(Ev E v nd d µ µ = =As we know that mobility of electrons. Drift Current The moment of electron in the presence of electric field.
  41. 41. Substitute the drift velocity value in equation 1 EnedriftJ ne nn n µ µσ = = )(
  42. 42. In case of semiconductor, the drift current density due to holes is given by eEpdriftJ pP µ=)( Then the total drift current density )()()( driftJdriftJdriftJ pn +=
  43. 43. eEpeEn pn µµ += pn pn epen E driftJ drift pneEdriftJ µµσ µµ +== += )( )( )()( For an intrinsic Semiconductor, n = p = ni, then )()( pnii endrift µµσ +=
  44. 44. Diffusion: Due to non-uniform carrier concentration in a semiconductor, the charge carriers moves from a region of higher concentration to a region of lower concentration. This process is known as diffusion of charge carriers.
  45. 45. Diffusion of charge carriers Drifting of charge carriers x Diffusion of charge carriers in a Semiconductor
  46. 46. Let Δn be the excess of electron concentration. Then according to Fick’s law, the rate of diffusion of electrons x n D x n n ∂ ∆∂ −= ∂ ∆∂− ∝ )( )(
  47. 47. )( )]([ n x eD n x De n n ∆ ∂ ∂ = ∆ ∂ ∂ −−= The diffusion current density due to holes )]([)( p x DediffusionJ pP ∆ ∂ ∂ −+= )( p x eDp ∆ ∂ ∂ −= Where Dn is the diffusion of electrons, the diffusion current density due to electrons is given by Jn(diffusion)
  48. 48. The total current density due to electrons is the sum of the current densities due to drift and diffusion of electrons )()( diffusionJdriftJJ nnn += )( )( p x eDEpeJ Similarly n x eDEneJ ppp nnn ∆ ∂ ∂ −= ∆ ∂ ∂ += µ µ
  49. 49. 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
  50. 50. Valence band Conduction band gE k E k E gE Valence band Conduction band
  51. 51. • 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
  52. 52. 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 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.
  53. 53. i B X Y Z VH + - _ _ _ __ _ _ __ __ _ _ _ _ _ _ _ P Q N – type Semiconductor
  54. 54. Magnetic deflecting force citydrift veloisvWhere )( )( d BvE qEBvq dH Hd ×= =× Hall eclectic deflecting force HqEF = When an equilibrium is reached, the Magnetic deflecting force on the charge carriers are balanced by the electric forces due to electric Field. )( BvqF d ×=
  55. 55. ne J vd = The relation between current density and drift velocit Where n is the number of charge carriers per unit volu BJ E ne tcoefficienHallR BJRE BJ ne E B ne J E BvE H H HH H H dH × ⇒= ×= ×= ×= ×= 1 ),.( )( ) 1 ( )( )(
  56. 56. be the Hall Voltage in equilibrium , the Hall Electric IB LV R Bd A I RV A I J JBdRV d V JB R JB E R d V E H H HH HH H H H H H H = = = = ×= = = sample,theofthicknesstheisLIf )( densitycurrentareasectionalcrossisAIf 1 slab.theofwidththeisdWhere
  57. 57. • Since all the three quantities EH , J and B are Measurable, the Hall coefficient RH and hence the carrier density can be find 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.

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