Nanostructures – density of states Faculty of Physics UW [email protected] Local density of states Gęstość stanów (ogólnie) można zdefiniować jako: 푁 퐸 = ෍ 훿 퐸 − 휀푛 푛 Jak widać po scałkowaniu: 퐸2 퐸2 퐸2 න 푁 퐸 푑퐸 = න ෍ 훿 퐸 − 휀푛 푑퐸 = ෍ න 훿 퐸 − 휀푛 푑퐸 퐸1 퐸1 푛 푛 퐸1 Przykładowo: 1 1 1 2푚 푁1퐷 퐸 = ෍ 훿 퐸 − 휀 푘 = න 훿 푘 − 푘′ 2 푑푘 = 2π 퐸′ 푘 휋 퐸 푘 1 1 푚 푁2퐷 퐸 = ෍ 훿 퐸 − 휀 푘 = න 훿 푘 − 푘′ 2휋푘 푑푘 = 2π 2 퐸′ 푘 휋ℏ2 푘 3/2 1 1 1 2푚 푁3퐷 퐸 = ෍ 훿 퐸 − 휀 푘 = න 훿 푘 − 푘′ 4휋푘2 푑푘 = 퐸 2π 3 퐸′ 푘 2휋2 ℏ2 푘 2017-06-05 2 Electrons statistics in crystals The case of a semiconductor, in which both the electron gas and hole gas are far from the degeneracy: the probability of filling of the electronic states: 퐸 퐸 휉 − 퐺 − 푒 + 퐸퐺 푓 ≈ 푒 2푘퐵푇 푘퐵푇 푘퐵푇 퐸 = + 퐸 퐸푒 푒 2 푒 퐸푐 and of holes 푓ℎ = 1 − 푓푒 퐸 퐸 휉 − 퐺 − ℎ − 2푘 푇 푘 푇 푘 푇 푓ℎ ≈ 푒 퐵 퐵 퐵 ∞ 휋 න 푥푒−푥 푑푥 = 2 0 퐸푣 퐸퐺 Thus: 퐸 = − − 퐸ℎ 퐸ℎ ∗ 3/2 퐸 휉 − 퐸 −휉 2 푚푒푘퐵푇 − 퐺 푐 푛 휉 = 2 푒 2푘퐵푇 ⋅ 푒푘퐵푇 = 푁 푇 푒 푘퐵푇 2휋ℏ2 푐 ∗ 3/2 퐸 휉 − 휉−퐸 푚ℎ푘퐵푇 − 퐺 − 푣 푝 휉 = 2 푒 2푘퐵푇 ⋅ 푒 푘퐵푇 = 푁 푇 푒 푘퐵푇 2휋ℏ2 푣 2017-06-05 3 Electrons statistics in crystals What is the concentration of carriers for T>0? In the thermodynamic equilibrium for an intrinsic semiconductors (półprzewodniki samoistne), the concentration of electrons in the conduction band is equal to the concnetration of holes in the valence band (because they appear only as a result of excitation from the valence band). 푛 = 푝 = 푛푖 (an intrinsic case) ln(n) 3 3 퐸 퐸 2 푘퐵푇 ∗ − − 푛 ⋅ 푝 = 푛 = 4 푚∗푚 2 푒 푘퐵푇 = 푁 푁 푒 푘퐵푇 푖 2휋ℏ2 푒 ℎ 푐 푣 3 2 3 퐸 퐸 푘퐵푇 ∗ − − 푛 = 푝 = 푛 = 2 푚∗ 푚 4 푒 2푘퐵푇 = 푁 푁 푒 2푘퐵푇 푖 2휋ℏ2 푒 ℎ 푐 푣 퐸 − 푒 2푘퐵푇 1/T 2017-06-05 4 Intrinsic carrier concentration What is the concentration of carriers for T>0? In the thermodynamic equilibrium for an intrinsic semiconductors (półprzewodniki samoistne), the concentration of electrons in the conduction band is equal to the concnetration of holes in the valence band (because they appear only as a result of excitation from the valence band). 3 3 퐸 퐸 2 푘퐵푇 ∗ ∗ − − 푛 ⋅ 푝 = 푛 = 4 푚 푚 2 푒 푘퐵푇 = 푁 푁 푒 푘퐵푇 band 푖 2휋ℏ2 푒 ℎ 푐 푣 (general formula) mpty e Eg 푛 = 푝 = 푛 푖 (an intrinsic3case) 2 3 퐸 퐸 푘퐵푇 ∗ − − 푛 = 푝 = 푛 = 2 푚∗ 푚 4 푒 2푘퐵푇 = 푁 푁 푒 2푘퐵푇 푖 2휋ℏ2 푒 ℎ 푐 푣 2휉−퐸 ∗ T 푁푐 1 3 푚ℎ 푘퐵푇 = 푒 ⇒ 휉 = 퐸푐 + 퐸푣 + 푘퐵푇 ln ∗ 푁푣 2 4 푚푒 in our notation the middle band of the band is 0 filled 2017-06-05 5 Dopants, impurities and defects Hydrogen-like model – ionization of the dopant n-type p-type conduction band conduction band energy donor level acceptor level Electron vallence band vallence band x x 2017-06-05 6 Dopants, impurities and defects Doping The carrier concentration in extrinsic semiconductor (niesamoistny) Consider a semiconductor, in which: NA – concentration of acceptors conduction band ED ND – concentration of donors Eg/2 pA – concentration of neutral acceptors nD – concentration of neutral donors energy donor level nc – concentration of electrons in conduction E 0 F band pv – concentration of holes in valence band Electron acceptor level E A From the charge neutrality of the crystal: -Eg/2 vallence band nc +(NA - pA)= pv + (ND - nD) nc + nD = (ND - NA)+ pv + pA x 2017-06-05 7 The occupation of impurity levels Occupation of impurity / defect levels in the thermodynamic equilibrium „Occupation” of localized or band states means the exchange of particles (electrons) between the reservoir and considered subsystem (microstate). The grand canonical ensemble (subsystem exchange particles and energy with the environment) Thermodynamic probability (unnormalized) of finding subsystem in a state 푗,in which there are 푛푗 particles (electrons) and which subsystem energy is 퐸푗 (the total energy of all 푛푗 particles): 1 −훽 퐸푗−푛푗휉 푃푗 = 푒 , 훽 = 푘퐵푇 휉- chemical potential. Statistical sum: −훽 퐸푗−푛푗휉 푍 = ෍ 푃푗 = ෍ 푒 푗 푗 2017-06-05 8 The occupation of impurity levels Statistical sum : −훽 퐸푗−푛푗휉 푍 = ෍ 푃푗 = ෍ 푒 푗 푗 Statistical means: −훽 퐸푗−푛푗휉 σ푗 퐴푗 ⋅ 푒 퐴 = −훽 퐸푗−푛푗휉 σ푗 푒 Examples: free electron occupying (or not) the quantum state of the 푘-vector and spin: 2 possible states of the subsystem (microstate): 푛0 = 0; 퐸0 = 0 푛1 = 1; 퐸1 = 퐸 (the occupation only for one spin state) the average number of particles of the subsystem: 0 ⋅ 푒0 + 1 ⋅ 푒−훽 퐸−휉 1 푛 = = = 푓 퐸, 푇 1 + 푒−훽 퐸−휉 푒훽 퐸−휉 + 1 (Fermi-Dirac distribution) 2017-06-05 9 The occupation of impurity levels Examples : free electron occupying (or not) the quantum state of the 푘-vector and ANY spin: 4 possible states of the subsystem (microstate): 푛0 = 0; 퐸0 = 0 푛1 = 1; 퐸1 = 퐸 (spin ↑) 푛2 = 1; 퐸2 = 퐸 (spin ↓) 푛3 = 2; 퐸3 = 2퐸 (spin ↑↓) the average number of particles of the subsystem: 0 ⋅ 푒0 + 1 ⋅ 푒−훽 퐸−휉 + 1 ⋅ 푒−훽 퐸−휉 + 2 ⋅ 푒−훽 2퐸−2휉 푒−훽 퐸−휉 1 + 푒−훽 퐸−휉 푛 = = 2 = 1 + 푒−훽 퐸−휉 + 푒−훽 퐸−휉 + 푒−훽 2퐸−2휉 푒−훽 퐸−휉 + 1 2 = 2푓 퐸, 푇 (Fermi-Dirac distribution x2) 2017-06-05 10 The occupation of impurity levels The ratio of the probability of finding dopant / defect of 푛 + 1 electrons and of 푛 electrons: σ 푒−훽 퐸푗− 푛+1 휉 푝푛+1 푁푛+1/푁푡표푡푎푙 푗:푛푗=푛+1 푔푛+1 = = = ⋅ 푒−훽 퐸푛+1−퐸푛 −휉 푝 푁 /푁 σ 푒−훽 퐸푘−푛휉 푔 푛 푛 푡표푡푎푙 푘;푛푘=푛 푛 σ푛 푁푛 = 푁 – impurity (dopants) concentration 퐸푛+1 i 퐸푛 – the lowest of all subsystem energies 퐸푗 with 푛 + 1 and 푛 electrons respectively Successive impurity energy levels are filled with the increase of the Fermi level. 퐸푛+1/푛 – so-called energy level of the impurity/ defect „numbered” by charge states 푛 + 1 and 푛 푔푛+1, 푔푛 – so-called degeneration of states of subsystem of 푛 + 1 and 푛 electrons 6/5/2017 11 The occupation of impurity levels 푔푛+1, 푔푛 – so-called degeneration of states of subsystem of 푛 + 1 and 푛 electrons. The degeneracy 푔푛+1 and 푔푛 takes into account the possibility of the existence of different subsystem states corresponding to a same number of particles (including the excited states): −훽휀푛,푖 푔푛 = 훼푛,0 + ෍ 훼푛,푖푒 푖=1,2,… 훼푛,0 i 훼푛,푖 they are respectively: the degeneracies of the 푛-electronic ground state and its excited states of energies higher than the ground state by 휀푛,푖 (excitation energies) Such defined degeneration 푔푛 generally depends on temperature 6/5/2017 12 The occupation of impurity levels Example – donor (we omit the excited states) charge state (+) implemented in 1 way : 푔+ = 1 charge state(0) implemented in 2 ways (spin ↑ or ↓): 푔0 = 2 0/+ the energy of the donor state Ε = 퐸퐷 Doping concentration of donors 푁퐷 푝푛+1 푝0 푔0 −훽 퐸퐷−휉 = = ⋅ 푒 , 푝+ + 푝0 = 1 푝푛 푝+ 푔+ The occupation probability of the donor state: 1 1 푝0 = 푛 = = 푔+ 훽 퐸 −휉 1 1 + ⋅ 푒 퐷 1 + ⋅ 푒훽 퐸퐷−휉 푔0 2 0 The concentration of occupied donor states (neutral donors are 푁퐷) 푁 푁0 = 퐷 퐷 1 1 + ⋅ 푒훽 퐸퐷−휉 TUTAJ 2017.04.10 2 2017-06-05 13 Dopants, impurities and defects Doping The carrier concentration in extrinsic semiconductor (niesamoistny) Consider a semiconductor, in which: NA – concentration of acceptors conduction band ED ND – concentration of donors Eg/2 pA – concentration of neutral acceptors nD – concentration of neutral donors energy donor level nc – concentration of electrons in conduction E 0 F band pv – concentration of holes in valence band Electron acceptor level E A From the charge neutrality of the crystal: -Eg/2 vallence band nc +(NA - pA)= pv + (ND - nD) nc + nD = (ND - NA)+ pv + pA x 2017-06-05 14 The occupation of impurity levels If both the donors and acceptors are shallow, and the electron and hole gas are not 0 0 degenerated, then : 퐸퐷 − 휉 ≫ 푘퐵푇, 휉 − 퐸퐴 ≫ 푘퐵푇, 푁퐷 ≪ 푁퐷, 푁퐴 ≪ 푁퐴 (that is, virtually all impurities are ionized) . 1 푛 = Δ푛 2 + 4푛2 + Δ푛 Δ푛 = 푛 − 푝 ≈ 푁 − 푁 2 푖 퐷 퐴 ⇒ 푛 ⋅ 푝 = 푛2 1 푖 푝 = Δ푛 2 + 4푛2 − Δ푛 2 푖 If Δ푛 > 0 (푛-type semiconductor – for 푝-type is the same) and Δ푛 ≫ 푛푖, (at 푇=300K: 푛푖(Ge) < 13 -3 11 -3 10 -3 10 cm , 푛푖(Si) < 10 cm , 푛푖(GaAs) < 10 cm ): 푛 ≈ 푁퐷 − 푁퐴 2 퐸 푛푖 푁퐶 푇 ⋅ 푁푉 푇 − 푝 ≈ = ⋅ 푒 푘퐵푇 푁퐷 − 푁퐴 푁퐷 − 푁퐴 majority carrier concentration is determined by the effective concentration of impurities, the concentration of minority carriers may be very small (e.g. Si). 2017-06-05 15 The occupation of impurity levels Compensation compensated semiconductors - containing both donors and acceptors At appropriately high temperatures majority carrier concentration given by the effective concentration of dopants |푁퐷 − 푁퐴| the concentration of scattering centers (charges): 푁퐷 + 푁퐴 compensation ratio - the ratio of impurity concentration: minority to majority.