Materials Science and Engineering - Electronic and Mechanical Properties of Materials Part 5 ppt

3.225 11 Density of Thermally Promoted of Carriers ∫ ∞ = c E dEEgEfn )()( Density of electron states per volume per dE Fraction of states occupied at a particular temperature Number of electrons per volume in conduction band ( dEeEEe m n Tk E E g Tk E e b g b F − ∞ ∫ −         = 2 1 2 3 2 * 2 2 2 1 h π Since 2 0 2 1 π = − ∞ ∫ dxex x , then Tk E Tk E be b g b F ee Tkm n −         = 2 3 2 * 2 2 h π N C ( ( TkEEe e Ef bF Tk EE Tk EE b F b F >>−≈ + = − − − )(when 1 1 )( Tk EE C b gF eNn − = ( 2 1 2 3 2 * 2 2 2 1 )( g e c EE m Eg −         = hπ © E. Fitzgerald-1999 ) ) ) ) 3.225 12 • A similar derivation can be done for holes, except the density of states for holes is used • Even though we know that n=p, we will derive a separate expression anyway since it will be useful in deriving other expressions Density of Thermally Promoted of Carriers ( 2 1 2 3 2 * 2 2 2 1 )( E m Eg h v −         = hπ )(1where,)()( 0 EffdEEgEfp hvh −== ∫ ∞− Tk E bh b F e Tkm p −         = 2 3 2 * 2 2 h π Tk E v b F eNp − = © E. Fitzgerald-1999 ) 6 3.225 13 Thermal Promotion • Because electron-hole pairs are generated, the Fermi level is approximately in the middle of the band gap • The law of mass action describes the electron and hole populations, since the total number of electron states is fixed in the system         +== * * ln 4 3 2 gives e h b g F m m Tk E Epn Since m e * and m h * are close and in the ln term, the Fermi level sits about in the center of the band gap ( Tk E ve b i b g emm Tk nnp 2 4 3 ** 2 3 2 2 2or −       == h π © E. Fitzgerald-1999 ) 3.225 14 Law of Mass Action for Carrier Promotion ( Tk E he b i b g emm Tk npn −       == 2 3 ** 3 2 2 2 4 h π Tk E VCi b g eNNn − = 2 ; • Note that re-arranging the right equation leads to an expression similar to a chemical reaction, where E g is the barrier. • N C N V is the density of the reactants, and n and p are the products. • + ′ → heNN g E VC VC i Tk E VC NN n e NN np b g 2 == − • Thus, a method of changing the electron or hole population without increasing the population of the other carrier will lead to a dominant carrier type in the material. • Photon absorption and thermal excitation produce only pairs of carriers: intrinsic semiconductor. • Increasing one carrier concentration without the other can only be achieved with impurities, also called doping: extrinsic semiconductors. © E. Fitzgerald-1999 ) 7 3.225 15 Intrinsic Semiconductors • Conductivity at any temperature is determined mostly by the size of the band gap • All intrinsic semiconductors are insulating at very low temperatures * 2 * 2 h h e e he m pe m ne pene ττ µµσ =+= Recall: ( Tk E hei b g een 2 int − ∝+= µµσ For Si, Eg=1.1eV, and let µ e and µ h be approximately equal at 1000cm 2 /V-sec (very good Si!). σ~10 10 cm -3 *1.602x10 -19 *1000cm 2 /V-sec=1.6x10 -6 S/m, or a resistivity ρ of about 10 6 ohm-m max. • One important note: No matter how pure Si is, the material will always be a poor insulator at room T. • As more analog wireless applications are brought on Si, this is a major issue for system-on-chip applications. This can be a measurement for E g © E. Fitzgerald-1999 + ) 3.225 16 Extrinsic Semiconductors • Adding ‘correct’ impurities can lead to controlled domination of one carrier type – n-type is dominated by electrons – p-type if dominated by holes • Adding other impurities can degrade electrical properties Impurities with close electronic structure to host Impurities with very different electronic structure to host isoelectronic hydrogenic x x x x Ge Si P x x x x x x x x Au Si deep level E c E v E c E v E c E v E D E DEEP - + © E. Fitzgerald-1999 8 3.225 17 Hydrogenic Model • For hydrogenic donors or acceptors, we can think of the electron or hole, respectively, as an orbiting electron around a net fixed charge • We can estimate the energy to free the carrier into the conduction band or valence band by using a modified expression for the energy of an electron in the H atom 2222 4 6.13 8 nnh me E o n −== ε 2 * 22222 4* 222 4 16.131 88 2 2 εεεε ε m m nnh em nh me E ro e e o n r −=→= = (in eV) • Thus, for the ground state n=1, we can see already that since ε is on the order of 10, the binding energy of the carrier to the center is <0.1eV • Expect that many carriers are then thermalize at room T • Experiment: • B acceptor in Si: .046 eV • P donor in Si: 0.044 eV • As donor in Si: 0.049 © E. Fitzgerald-1999 3.225 18 The Power of Doping • Can make the material n-type or p-type: Hydrogenic impurities are nearly fully ionized at room temperature –n i 2 for Si: ~10 20 cm -3 – Add 10 18 cm -3 donors to Si: n~N d –n~10 18 cm -3 , p~10 2 (n i 2 /N d ) • Can change conductivity drastically – 1 part in 10 7 impurity in a crystal (~10 22 cm -3 atom density) –10 22 *1/10 7 =10 15 dopant atoms per cm -3 –n~10 15 , p~10 20 /10 15 ~10 5 σ/σ i ~(p+n)/2n i ~n/2n i ~10 5 ! Impurities at the ppm level drastically change the conductivity (5-6 orders of magnitude) © E. Fitzgerald-1999 9 3.225 19 Expected Temperature Behavior of Doped Material (Example:n-type) • 3 temperature regimes ln(n) 1/T Intrinsic Extrinsic Freeze-out E g /2k b E b /k b © E. Fitzgerald-1999 3.225 20 Contrasting Semiconductor and Metal Conductivity • Semiconductors – changes in n(T) can dominate over τ – as T increases, conductivity increases • Metals – n fixed – as T increases, τ decreases, and conductivity decreases σ τ = ne m 2 © E. Fitzgerald-1999 10 3.225 21 General Interpretation of τ • Metals and majority carriers in semiconductors – τ is the scattering length – Phonons (lattice vibrations), impurities, dislocations, and grain boundaries can decrease τ 11111 ++++= gbdislimpurphonon τττττ 1 1 = == iii iithth i i Nl Nvv l σ σ τ where σ is the cross-section of the scatterer, N is the number of scatterers per volume, and l is the average distance before collisions The mechanism that will tend to dominate the scattering will be the mechanism with the shortest l (most numerous), unless there is a large difference in the cross-sections Example: Si transistor, τ phonon dominates even though τ impur gets worse with scaling. © E. Fitzgerald-1999 3.225 22 Estimate of T dependence of conductivity • τ ~l for metals • τ ~l/v th for semiconductors • First need to estimate l=1/N σ 2 1 x N l ion ionion ph πσ σ ∝ = x=0 ∫ ∫ ∞+ ∞− +∞ ∞− ΨΨ ΨΨ = dx dxx x * 2* 2 Use Ψ for harmonic oscillator, get: 1 2 − == kT e Exk ω ω h h Average energy of harmonic oscillator © E. Fitzgerald-1999 11 3.225 23 Estimate of T dependence of conductivity 1 1 2 − = = − == T kT e k E k e Exk θ ω θ θω ω h h h Therefore, <x 2 > is proportional to T if T large compared to θ: TNv xNv v l T x l Tx T e ionF ionF F cond T π π τµσ σ θ θ 11 111 1 2 2 2 ==≈∝∝ ∝∝∝ ∝ +≈ For a metal: For a semiconductor, remember that the carriers at the band edges are classical-like: 2 3 2 1 * 1 3 − ∝∝== T T T m kT l v l th τ 2 3 * − ∝= T m eτ µ © E. Fitzgerald-1999 3.225 24 Example: Electron Mobility in Ge µ~T -3/2 if phonon dominated (T -1/2 from vth, T -1 from x-section σ At higher doping, the ionized donors are the dominate scattering mechanism © E. Fitzgerald-1999 12 1 3.225 1 Minority Carrier Lifetimes, τ • Minority carriers (e.g. electrons (minority carrier) in p-type material with majority holes τ is the time to recombination: recombination time – means for system to return to equilibrium after perturbation, e.g. by illumination E c E v τ, l Recombination x E © E. Fitzgerald-1999 Generation • Deep levels in semiconductors act as carrier traps and/or enhanced recombination sites E c E v Recombination through deep level E deep 3.225 2 Generation and Recombination • Generation – photon-induced or thermally induced, G=#carriers/vol sec – e.g. g = P/hν –G o is the equilibrium generation rate • Recombination – R=# carriers/vol sec –R o is the equilibrium recombination rate, balanced by G o • Net change in carrier density: – dn/dt = G - R = G - (n - n 0 ) / τ = G - ∆ n / τ – Under steady state illumination: dn/dt = 0 –n p (0) = n p0 + G τ Αfter turning off illumination: –n p (0) = n p0 + G τ e -t/τ © H.L. Tuller, 2001 g τ t n p (t) n p0 n p (0) τ 3.225 3 Key Processes: Drift and Diffusion Electric Field: Drift Concentration Gradient: Diffusion EenJAenvI EepJAepvI eede hhdh µ µ == == ; ; neDJ peDJ ee hh ∇= ∇−= neDEenJ peDEepJ eeeTOT hhhTOT ∇+= ∇−= µ µ © E. Fitzgerald-1999 3.225 4 Electrochemical Potential ϕµη qz jjj += jjj ckT ln 0 +=µµ = ϕ xqz j j j j j ∂ ∂         − = ησ x c qDz x j jjj ∂ ∂ − ∂ ∂ = ϕ σ Note: Under equilibrium conditions 0= ∂ ∂ x j η Electrochemical Potential ⇒E F Chemical Potential Electrostatic Potential © H.L. Tuller, 2001 2 3.225 5 Continuity Equations • For a given volume, change in carrier concentration in time is related to J GR TOT GR TOT GRdiffdrift t p t p J et p t n t n J et n t n t n t n t n t n ∂ ∂ + ∂ ∂ −⋅∇−= ∂ ∂ ∂ ∂ + ∂ ∂ −⋅∇= ∂ ∂ ∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ = ∂ ∂ 1 1 1-D, GR x p D x E p t p GR x n D x E n t n hh ee +− ∂ ∂ + ∂ ∂ −= ∂ ∂ +− ∂ ∂ + ∂ ∂ = ∂ ∂ 2 2 2 2 µ µ © E. Fitzgerald-1999 3.225 6 Minority Carrier Diffusion Equations • In many devices, carrier action outside E-field controls properties > minority carrier devices • Only diffusion in these regions e h h e n R p R GR x p D t p GR x n D t n τ τ ∆ −= ∆ −= +− ∂ ∂ = ∂ ∂ +− ∂ ∂ = ∂ ∂ type,-pin type,-nin 2 2 2 2 Assuming low-level injection, t n t n t n t n o ∂ ∆∂ ≈ ∂ ∆∂ + ∂ ∂ = ∂ ∂ therefore materialtype-nin materialtype-pinG 2 2 2 2 G p x p D t p n x n D t n h h e e + ∆ − ∂ ∆∂ = ∂ ∆∂ + ∆ − ∂ ∆∂ = ∂ ∆∂ τ τ © E. Fitzgerald-1999 3 . Eg=1.1eV, and let µ e and µ h be approximately equal at 1000cm 2 /V-sec (very good Si!). σ~10 10 cm -3 *1.602x10 -1 9 *1000cm 2 /V-sec=1.6x10 -6 S/m, or a resistivity ρ of about 10 6 ohm-m max –10 22 *1/10 7 =10 15 dopant atoms per cm -3 –n~10 15 , p~10 20 /10 15 ~10 5 σ/σ i ~(p+n)/2n i ~n/2n i ~10 5 ! Impurities at the ppm level drastically change the conductivity ( 5- 6 orders of magnitude). 3.2 25 11 Density of Thermally Promoted of Carriers ∫ ∞ = c E dEEgEfn )()( Density of electron states per volume per dE Fraction of states occupied at a particular temperature Number of