Itroductio to Solid State Physics Class: Itegrated Photoic Devices Time: Fri. 8:00am ~ 11:00am. Classroom: 資電 206 Lecturer: Prof. 李明昌 (Mig-Chag Lee) Electros i A Atom
Electros i A Atom Electros i Two atoms
Aalogy to A Coupled Two-Pedulum System f f f 1 sprig f 2 Oe-dimesioal Kroig-Peey Model Electro Potetial Eergy Potetial Eergy Electro
Solutios of the S.E. i Kroig-Pey Model Periodic Potetial Well a+b Schrödiger s equatio 2 2 Ψ + 2m ( U ( x) E) Ψ = 0 solutio Bloch Theorem If The U [ x + ( a + b)] = U ( x) Ψ( x) = exp( jk x) u( x) where u [ x + ( a + b)] = u( x) E-k diagram Desity of States as a fuctio of k ad E For electros ear the bottom of the bad, the bad itself form a pseudo-potetial well. The well bottom lies at E c ad the termiatio of the bad at the crystal surfaces forms the walls of the well
Desity of States as a fuctio of k ad E The desity of states ear the bad edges ca therefore be equal to the desity of states available to a particle of mass m * i a three dimesio box with the dimesio of the crystal U( x ) = 0 z except boudary ϕ ϕ ϕ x y z (Time-idepedet S.E.) 2 2 2 2 + + + k ϕ = 0 2 2 2 (a) 0 < x < a, 0 < y< b, 0< z< c k or 2 me/ k E = 2m 2 2 2 Boudary Desity of States as a fuctio of k ad E To solve the equatio, we employ the separatio of variables techique; that is Substitute (b) to (a) 2 1 d ϕx = costat = k 2 ϕ dx x ϕ( x, yz, ) = ϕ ( x) ϕ ( y) ϕ ( z) x y z 1 1 d ϕ 1 ϕ ϕ ϕ 2 2 2 d ϕx y d ϕz 2 + + + k = 2 2 2 x dx y dy x dz where 2 x k = k + k + k 2 2 2 2 x y z 0 ϕ ( x, yz, ) = Asi kx si ky si kz x y z xπ yπ zπ kx = ; ky = ; kz = x, y, z =± 1, ± 2, ± 3,... a b c (b) (Solutio)
Desity of States as a fuctio of k ad E - Desities of states i k-space A uit cell of volume: π π π ( )( )( ) a b c The larger the a,b ad c, the smaller the uit cell volume Desity of States as a fuctio of k ad E How may states are withi k ad k+dk? spi up ad dow Eergy states with k abc 2 1 σ( kdk ) = (4 πk) 2 dk 3 betwee k ad k + dk π 8 Desity of states Redudacy
Desity of States as a fuctio of k ad E How may states are withi E ad E+dE? σ( EdE ) = σ( kdk ) σ( E) = σ( k) dk de From the previous equatio, k E = 2m 2 2 de dk 2 k = 2 ( ) ( ) m me Therefore σ E = abc 2 3 m π Desity of states per uit volume m 2mE m 2mE ρ( E) = σ( E)/ V = ( abc) / V = 2 3 2 3 π π V = abc Electros ad Holes i Semicoductors Electric Field
Electro Distributio Fuctios The probability desity fuctio of electro residig i eergy level E ca be represeted by Fermi-Dirac distributio 1 f ( E) = E E 1+ exp kt F where E F : Fermi Eergy Level Carrier Desity Distributio of Itrisic Semicoductors =
Metal, Isulator, ad Semicoductor Metal Metal Isulator Semicoductor Work Fuctio Work fuctio is the miimum eergy required to eable a electro to escape from the surface of solid. Work fuctio φis the eergy differece betwee Fermi level ad the vacuum level. I semicoductors, it is more usual to use the electro affiity χ, defied as the eergy differece betwee the bottom of the coductio bad ad the vacuum level.
Extrisic Semicoductors Carrier Desity Distributio of Extrisic Semicoductors = -type p-type
p- Juctio (a). Iitially separated p-type ad -type semicoductor; (b) the eergy bad distortio after the juctio is formed; (c) the space charge layers of ioized impurity atoms withi the depletio regio W; ad (d) the potetial distributio at the juctio. Cotact Potetial o p- juctio Cotact potetial V 0 The electro cocetratio i the coductio bad of p-type side as EC E p F p p = Ncexp kt Similarly the electro cocetratio i the -type side is EC E F = Ncexp kt Sice the Fermi level is costat. EC E l p C = kt = ev p 0 E = E = E F F F V p kt = e p 0 l
Miority Carrier Distributio At the temperature i the rage of 100K T 400K, the majority carrier cocetratios are equal to the dopig levels, that is Pp = N ad = Nd, kt NN a d 2 V0 = l (recall p= i ) e Carrier cocetratio differece betwee p- ad -type semicoductor ad p ev0 = exp kt i a p ev0 = ppexp kt Curret flow i a forward-biased p- juctio The juctio is said to be forward biased if the p regio is coected to the positive termial of the voltage source The exteral voltage is dropped across the depletio regio ad lower dow the potetial barrier by (V 0 -V). So the diffusio curret becomes larger tha the drift curret. There is a et curret from the p to the regio. The Fermi levels are o loger aliged across the juctio i light of the exteral voltage.
Curret flow i a forward-biased p- juctio Carrier desities Oce the majority carriers flow across depletio regio, they become miority carriers. The miority cocetratios ear the juctio rise to ew value p ad p. The majority carrier cocetratio is almost uchaged uless a large curret ijectio Due to the miority cocetratio gradiet, the ijected curret diffuses away from the juctio. The oliear gradiet idicates miority holes (electros) are recombied with electros (holes) that are repleished by exteral voltage source. Curret flow i a forward-biased p- juctio The ijected carrier cocetratio at the edge of depletio regio Sice p p ev ( 0 V) ' = exp kt ev ( 0 V) p' = ppexp kt ev0 = ppexp kt ad p ev = 0 exp kt The p ' = ev p exp kt ad ' = p p ev exp kt (a)
Curret flow i a forward-biased p- juctio As we oted i previous slides, the excess miority carrier cocetratio will decrease due to recombiatio x px ( ) = p(0)exp( ) L where p( x) = p '( x) p h Therefore, from (a) ev p(0) = p exp 1 kt Curret flow i a forward-biased p- juctio The diffusio curret At x = 0 J J h h edh x = p(0)exp( ) L L h edh ev = p exp( ) 1 L kt h Similarly, for electro diffusio curret at the depletio edge The total curret J ev J = J0 exp 1 kt e ede ev = p exp( ) 1 L kt e where h D D J = e p + h e 0 p Lh Le
Curret flow i a reverse-biased p- juctio J ev = J0 exp 1 kt V is egative I-V curves of p- juctio I-V Curve J ev = J0 exp 1 kt Reversed Bias Forward Bias
Zeer Breakdow Tuellig of carriers across the depletio regio ad the process is idepedet of temperature Dopig desities must be high to occur before avalachig Breakdow curret is idepedet of voltage Metal-semicoductor Juctio --- Schottky Cotact φ m > φ s A schottky barrier formed by cotactig a metal to a -type semicoductor with the metal havig the larger work fuctio: bad diagrams (a) before ad (b) after the juctio is formed.
Work Fuctio of Metal Electro Affiity of Semicoductor
Metal-semicoductor Juctio --- Schottky Cotact The metal to -type semicoductor juctio show uder (a) forward bias ad (b) reverse bias. Metal-semicoductor Juctio --- Ohmic Cotact φ m < φ s
Summary of Metal-semicoductor Juctio Cotact Schottly p-n heterojuctio
-P heterojuctio Double Heterojuctios
Carriers Cofied i Double Heterojucitos It is extremely efficiet to ehace electro-hole recombiatio Fudametals of Electromagetic Waves Electrical field Electrical displacemet Magetic field E( r, t) D( r, t) H ( r, t) V/m C/m 2 A/m Magetic iductio B( r, t) webers/m 2 space time E ad B are fudametal fields D ad H are derived from the respose of the medium How may variables i Electromagetic Wave?