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![EE4353/5353 – Fundamentals of Advanced Semiconductor Technology
Carrier Mobility, m
mn [qtmn / mn*] is the electron mobility
mp [qtmp / mp*] is the hole mobility
Similarly, for holes:
Si Ge GaAs InAs
mn (cm2/Vs) 1400 3900 8500 30,000
mp (cm2/Vs) 450 1900 400 500
Electron and hole mobilities for intrinsic semiconductors @ 300K
For electrons:
𝜐𝑑𝑝 = ൘
𝑞ℇ𝜏𝑚𝑝
𝑚𝑝
∗ ≡ 𝜇𝑝ℇ
𝜐𝑑𝑛 = ൗ
𝑞ℇ𝜏𝑚𝑛
𝑚𝑛
∗ ≡ 𝜇𝑛ℇ](https://image.slidesharecdn.com/l-04-carrieraction-240416203525-77fbb3f7/75/The-carrier-structure-of-a-P-and-N-type-Semiconductor-7-2048.jpg)
The carrier structure of a P and N type Semiconductor
- 1. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Lecture 4 Carrier Action EE 4353/5353 Dr. Ariful Haque
- 2. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Carrier Action • The three primary action – drift: charged-particle motion in response to an applied electric field – diffusion: process whereby particles tend to spread out as a result of their difference of concentrations – recombination-generation: • Generation is a process whereby carriers are created • Recombination is a process whereby carriers are destroyed • Electrons in semiconductors have 3 degrees of freedom ➔ according to the equipartition theorem the kinetic energy of the electrons is: 1 2 𝑚𝑛𝑣𝑡ℎ 2 = 3 2 𝑘𝑇, where vth is the average thermal velocity
- 3. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Carrier Scattering • Mobile electrons and atoms in the Si lattice are always in random thermal motion. – Electrons make frequent collisions with the vibrating atoms “lattice scattering” or “phonon scattering” – increases with increasing T • Other scattering mechanisms: – deflection by ionized impurity atoms – deflection due to Coulombic force between carriers “carrier-carrier scattering” – only significant at high carrier concentrations • The net current in any direction is zero, if no E-field is applied. 1 2 3 4 5 electron
- 4. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Carrier Drift • When an electric field (e.g. due to an externally applied voltage) exists within a semiconductor, mobile charge-carriers will be accelerated by the electrostatic force: Electrons drift in the direction opposite to the E-field → net current Because of scattering, electrons in a semiconductor do not undergo constant acceleration. However, they can be viewed as quasi- classical particles moving at a constant average drift velocity vdn 𝐹 = (−𝑞)ℇ 𝜐𝑑𝑛 = 𝜇𝑛ℇ
- 5. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Conductivity and Mobility • Electric field in the +x direction • Each electron experiences force −qℰ𝑥 ⟹ net motion in the –x direction • Net force of the field on the n electrons/cm-3 is: • This net acceleration is balanced by decelerations of the collision processes −nqℰ𝑥 = ቤ 𝑑p𝑥 𝑑𝑡 𝑓𝑖𝑒𝑙𝑑 • With collisions included the net rate of change of momentum must be zero Differential change in 𝑝𝑥 due to collisions in time dt is: 𝑑p𝑥 = −p𝑥 𝑑𝑡 𝜏 𝜏 is the mean time between scattering events of the mean free time px= momentum in the x-direction
- 6. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Conductivity and Mobility Rate of change of 𝑝𝑥 due to decelerating effect of collisions is: ቤ 𝑑𝑝𝑥 𝑑𝑡 𝑐𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛𝑠 = − 𝑝𝑥 𝜏 For steady state the sum of accelerating and decelerating effects must be zero: The avg. momentum per electron: − 𝑝𝑥 𝜏 − nqℰ𝑥 = 0 𝑝𝑥 = 𝑝𝑥 𝑛 = −q𝜏ℰ𝑥 For steady state the electrons have a constant average velocity in the –x direction 𝑣𝑥 = 𝑝𝑥 𝑚𝑛 ∗ = − q𝜏 𝑚𝑛 ∗ ℰ𝑥 ---- represents the net drift of an electron in response to an electric field Net current density resulting from this drift is: 𝐽𝑥 = −𝑞𝑛 𝑣𝑥 = 𝑛𝑞2𝜏 𝑚𝑛 ∗ ℰ𝑥 = 𝜎ℰ𝑥 𝑤ℎ𝑒𝑟𝑒 𝜎 ≡ 𝑛𝑞2𝜏 𝑚𝑛 ∗ Conductivity can be written as: 𝜎 = 𝑞𝑛𝜇𝑛 where 𝜇𝑛 = 𝑞𝜏 𝑚𝑛 ∗ Mobility
- 7. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Carrier Mobility, m mn [qtmn / mn*] is the electron mobility mp [qtmp / mp*] is the hole mobility Similarly, for holes: Si Ge GaAs InAs mn (cm2/Vs) 1400 3900 8500 30,000 mp (cm2/Vs) 450 1900 400 500 Electron and hole mobilities for intrinsic semiconductors @ 300K For electrons: 𝜐𝑑𝑝 = ൘ 𝑞ℇ𝜏𝑚𝑝 𝑚𝑝 ∗ ≡ 𝜇𝑝ℇ 𝜐𝑑𝑛 = ൗ 𝑞ℇ𝜏𝑚𝑛 𝑚𝑛 ∗ ≡ 𝜇𝑛ℇ
- 8. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Drift Velocity Calculation a) Find the hole drift velocity in an intrinsic Si sample for ℇ = 103 V/cm. b) What is the average hole scattering time? (Use: 𝑚𝑝 ∗= 0.4𝑚𝑜, 𝜇𝑝 = 450 𝑐𝑚2𝑉−1𝑠−1) 𝑚𝑝 ∗ 𝜐𝑑𝑝 = 𝑞ℇ𝜏𝑚𝑝 ⟹ 𝜐𝑑𝑝 = 𝑞ℇ𝜏𝑚𝑝 𝑚𝑝 ∗ = 𝜇𝑝ℇ 𝜐𝑑𝑝 = 450𝑐𝑚2𝑉−1𝑠−1x103V𝑐𝑚−1 = 4.5𝑥105𝑐𝑚𝑠−1 𝜏𝑚𝑝 = 𝑚𝑝 ∗ 𝜇𝑝 𝑞 = 0.4𝑥9.1𝑥10−31 𝑘𝑔 𝑥(450𝑐𝑚2 𝑉−1 𝑠−1 ) 1.6𝑥10−19𝐶 = 10−9 𝑘𝑔𝑐𝑚2 𝑉−1 𝑠−1 𝐶 = 10−13 𝑠 = 0.1𝑝𝑠
- 9. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Conductivity and Resistivity • In a semiconductor, both electrons and holes conduct current: 𝐽𝑝,𝑑𝑟𝑖𝑓𝑡 = 𝑞𝑝𝜇𝑝ℰ 𝐽𝑛,𝑑𝑟𝑖𝑓𝑡 = −𝑞𝑛(−𝜇𝑛)ℰ 𝐽𝑑𝑟𝑖𝑓𝑡 = 𝐽𝑝,𝑑𝑟𝑖𝑓𝑡 + 𝐽𝑛,𝑑𝑟𝑖𝑓𝑡 = 𝑞𝑝𝜇𝑝ℰ + 𝑞𝑛𝜇𝑛ℰ 𝐽𝑑𝑟𝑖𝑓𝑡 = (𝑞𝑝𝜇𝑝 + 𝑞𝑛𝜇𝑛)ℰ • The conductivity of a semiconductor is 𝜎 ≡ 𝑞𝑝𝜇𝑝 + 𝑞𝑛𝜇𝑛 ( Τ 𝑚ℎ𝑜 𝑐𝑚) • The resistivity of a semiconductor is 𝜌 ≡ 1 𝜎 ( Τ 𝑜ℎ𝑚 𝑐𝑚)
- 10. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Drift and Resistance • Consider a semiconductor bar with both types of carrier • Knowing the conductivity, the resistance can be determined • The electric field causes the holes and electrons to drift in opposite directions • Both hole and electron current are in the direction of the electric field 𝑅 = 𝜌𝐿 𝑤𝑡 = 𝐿 𝑤𝑡 . 1 𝜎 • Drift current is constant throughout the bar • Contacts are ohmic: perfect source and sink of both type of carriers • Consider electron current in the circuit → for every electron leaving the bar at x=0, corresponding electron enter the bar at x=L →electron conc. in the bar is constant • What about holes? As hole reaches contact at x=L, it recombines with an electron supplied by external circuit. • This hole disappears but one must appear at x=0 to maintain space charge neutrality • Reasonable to consider the source of this hole as generation of EHP at x=0
- 11. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Mechanisms of Carrier Scattering Dominant scattering mechanisms: 1. Phonon scattering (lattice scattering) • Scattering by vibration of lattice • Increases with temperature • Decrease in mobility 2 / 3 2 / 1 1 velocity ermal carrier th density phonon 1 − T T T phonon phonon t m Phonon scattering limited mobility decreases with increasing T: m = qt / m T vth T µ
- 12. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology There is less change in the electron’s direction if the electron travels by the ion at a higher speed. Ion scattering limited mobility increases with increasing T: T µ 2. Impurity (dopant) ion scattering Mechanisms of Carrier Scattering • Dominant at low temperatures • Lattice scattering is reduced • Thermal motion of electrons is reduced 𝜇𝑖𝑚𝑝𝑢𝑟𝑖𝑡𝑦 ∝ 𝑣𝑡ℎ 3 𝑁𝐴 + 𝑁𝐷 ∝ 𝑇 Τ 3 2 𝑁𝐴 + 𝑁𝐷
- 13. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Mechanisms of Carrier Scattering • Scattering probability is inversely proportional to the mean free time and therefore the mobility. • Mobility due to 2 or more scattering mechanisms add inversely…. 1 𝜇 = 1 𝜇1 + 1 𝜇2 +….
- 14. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Matthiessen's Rule → Probability that a carrier will be scattered by any mechanism within a time period dt is σ𝑖 Τ 𝑑𝑡 𝜏𝑖 impurity phonon impurity phonon m m m t t t 1 1 1 1 1 1 + = + = • The probability that a carrier will be scattered by mechanism i within a time period dt is Τ 𝑑𝑡 𝜏𝑖 ti ≡ mean time between scattering events due to mechanism i The dominant scattering mechanism (with the lowest associated 𝜇𝑖) determines 𝜇 Since 𝜇 = Τ 𝑞𝜏 𝑚
- 15. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Mobility Dependence on Doping Carrier mobilities in Si at 300K Phonon scattering is dominant Ionized impurity scattering becomes dominant Screening effect becomes significant
- 16. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Mobility Dependence on Temperature impurity phonon m m m 1 1 1 + =
- 17. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Problem Calculate the mean free time of an electron having mobility of 1000 cm2/V-s at 300K. Also calculate the mean free path. Assume mn=0.26m0
- 18. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Problem Calculate the mean free time of an electron having mobility of 1000 cm2/V-s at 300K. Also calculate the mean free path. Assume mn=0.26m0 𝜏 = 𝑚𝑛𝜇𝑛 𝑞 = 0.26𝑥0.91𝑥10−30 𝑘𝑔 𝑥 1000𝑥10−4 Τ 𝑚2 𝑉 − 𝑠 1.6𝑥10−19𝐶 = 1.48𝑥10−13𝑠 = 0.148𝑝𝑠 Mean free time: Thermal velocity from: 1 2 𝑚𝑛𝑣𝑡ℎ 2 = 3 2 𝑘𝑇, 𝑣𝑡ℎ = 2.288𝑥107 Τ 𝑐𝑚 𝑠 Mean free path: 𝑙 = 𝑣𝑡ℎ𝜏 = 2.288𝑥107 Τ 𝑐𝑚 𝑠 1.48𝑥10−13 𝑠 = 33.7 𝑛𝑚
- 19. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology The Hall Effect • Hole drift in a p-type bar • Magnetic field applied perpendicular to hole drift direction • Path of the hole will be deflected • Total force on a single hole: • In the y-direction: 𝑭 = 𝑞(ℰ + 𝒗 𝑋 ℬ) 𝐹𝑦 = 𝑞(ℰ𝑦 − 𝑣𝑥𝐵𝑧) • Examining the equation: unless an electric field ℰ𝑦 is established along the width of the bar each hole will experience a net force along the –y direction due to the q𝑣𝑥𝐵𝑧 product. • To maintain a steady flow of holes along the bar, the electric field ℰ𝑦 must just balance the product 𝑣𝑥𝐵𝑧. ℰ𝑦 = 𝑣𝑥𝐵𝑧
- 20. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology The Hall Effect ℰ𝑦 = 𝑣𝑥𝐵𝑧 • The establishment of ℰ𝑦 is known as the Hall effect • The voltage, 𝑉𝐴𝐵 = ℰ𝑦𝑤 is called the Hall voltage • Using the equation for drift velocity, ℰ𝑦 = 𝐽𝑥 𝑞𝑝 𝐵𝑧 = 𝑅𝐻𝐽𝑥𝐵𝑧, 𝑅𝐻 ≡ 1 𝑞𝑝 • The Hall field is proportional to the current density and the magnetic field • The term RH is called the Hall coefficient • A measurement of the Hall voltage VAB, knowing the current and magnetic field can be used to determine the carrier concentration 𝑝 = 1 𝑞𝑅𝐻 = 𝐽𝑥𝐵𝑧 𝑞ℰ𝑦 = Τ 𝐼𝑥 𝑤𝑡 𝐵𝑧 𝑞 Τ 𝑉𝐴𝐵 𝑤 = 𝐼𝑥𝐵𝑧 𝑞𝑡𝑉𝐴𝐵
- 21. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology The Hall Effect 𝑝 = 1 𝑞𝑅𝐻 = 𝐽𝑥𝐵𝑧 𝑞ℰ𝑦 = Τ 𝐼𝑥 𝑤𝑡 𝐵𝑧 𝑞 Τ 𝑉𝐴𝐵 𝑤 = 𝐼𝑥𝐵𝑧 𝑞𝑡𝑉𝐴𝐵 Since all the quantities can be measured, the Hall effect can give accurate values for the carrier concentration • A measurement of the resistance can give the resistivity 𝜌 Ω − 𝑐𝑚 = 𝑅𝑤𝑡 𝐿 = Τ 𝑉𝐶𝐷 𝐼𝑥 Τ 𝐿 𝑤𝑡 But the conductivity, 𝜎 = 𝑞𝜇𝑝𝑝 𝜇𝑝 = 𝜎 𝑞𝑝 = Τ 1 𝜌 𝑞( Τ 1 𝑞𝑅𝐻) = 𝑅𝐻 𝜌 The mobility is the ratio of the Hall coefficient and the resistivity Sign of the Hall voltage give an indication whether the material is n-type or p-type
- 22. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Example w = 1mm t = 10 µm L = 5mm For B = 10kG (10-4 Wb/cm2) and current I =1mA, Voltage VAB = -2mV VCD = 100mV Charge type? majority carrier concentration and mobility?
- 23. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Example w = 1mm t = 10 µm L = 5mm For B = 10kG (10-4 Wb/cm2) and current I =1mA, Voltage VAB = -2mV VCD = 100mV Charge type? majority carrier concentration and mobility? 𝑛 = 𝐼𝑥𝐵𝑧 𝑞𝑡(−𝑉𝐴𝐵) = (10−3)(10−4) 1.6𝑥10−19 10−3 (2𝑥10−3) = 3.125𝑥1017 𝑐𝑚−3
- 24. EE4353/5353 – Fundamentalsof Advanced Semiconductor Technology Resistivity Dependence on Doping For n-type material: n qnm 1 For p-type material: p qpm 1 Note: This plot (for Si) does not apply to compensated material (doped with both acceptors and donors).