Complementi di Fisica Lecture 24

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1 Comlemeti di Fisica - Lecture Comlemeti di Fisica Lecture 24 Livio Laceri Uiversità di Trieste Trieste, I this lecture Cotets Drift of electros ad holes i ractice (umbers ): coductivity σ as a fuctio of imurity cocetratio bad bedig : reresetatio of macroscoic electric fields Diffusio i ractice (umbers ): Diffusio coefficiet D Iduced ( Built-i ) electric field Eistei relatio betwee σ ad D Referece tetbooks D.A. Neame, Semicoductor Physics ad Devices, McGraw- Hill, 3 rd ed., 2003, ( 5 Carrier Trasort Pheomea ) R.Pierret, Advaced Semicoductor Fudametal, Pretice Hall, 2d ed., ( 6 Carrier Trasort ) L.Laceri - Comlemeti di Fisica - Lecture 24 2 L.Laceri - Comlemeti di Fisica 1

2 Comlemeti di Fisica - Lecture From the Boltzma Euatio The cotiuity euatios for the electrical curret desity i semicoductors ca be obtaied from the Boltzma euatio: f t v g f F r " f f f 0 k τ F e E Force o electros Multilyig by the grou velocity ad itegratig over the mometum sace dk dk y dk z : v f d 3 k g t v g ( v g r f )d 3 k ' F v g ) " ( k f *, d 3 k + Oe obtais the eressio for curret desity (detailed derivatio: see FELD , MOUT ) f f v 0 g d 3 k τ L.Laceri - Comlemeti di Fisica - Lecture 24 3 Drift-diffusio cotiuity euatio τ J t For electros (similar for holes): + J % µ ' E + 1 & k B T r + k B r T ( * ) Drift Diffusio already discussed i the Drude model relaatio time τ is small: This ew term ca be eglected if freuecy is ot too high (few hudred MHz) Temerature gradiet: also a temerature gradiet ca drive a electric curret Abset at thermal euilibrium L.Laceri - Comlemeti di Fisica - Lecture 24 4 L.Laceri - Comlemeti di Fisica 2

3 Comlemeti di Fisica - Lecture D Drift ad diffusio: Eistei relatio Drift (mobility) ad diffusio (diffusivity) coefficiets are correlated thermal velocity D v th l v & kt # $ µ % " drift velocity µ E Why? See revious lecture Also: o curret at euilibrium, see et D v eteral field µ E & kt # $ µ % " L.Laceri - Comlemeti di Fisica - Lecture 24 5 Mobility ad diffusivity vs. cocetratio Low doig cocetratio: Lattice scatterig limit mobility µ (cm 2 V -1 s -1 ) T ~ 300 K High doig cocetratio: Imurity scatterig limit diffusivity D (cm 2 s -1 ) Electros ad holes: m µ > µ < m L.Laceri - Comlemeti di Fisica - Lecture 24 6 L.Laceri - Comlemeti di Fisica 3

4 Comlemeti di Fisica - Lecture Bad bedig : built-i electrical field J V No-uiform doig: built-i field -tye doig, thermal euilibrium (o eteral el.field alied): ( µ & Ɛ V ' Ɛ is the built-i electric field: 1 d 1 dn A Ɛ Vth Vth N th d µ Ɛ D 0 kt d µ Ɛ µ kt th 1 d % # 0 $ thermal voltage euivalet A () () diffusio -N - A - + Small sace charge ubalace Ɛ() Built-i field drift L.Laceri - Comlemeti di Fisica - Lecture 24 8 holes electros holes electros comesates diffusio L.Laceri - Comlemeti di Fisica 4

5 Comlemeti di Fisica - Lecture No-uiform doig i euilibrium Uder thermal ad diffusive euilibrium coditios the Fermi level ( total chemical otetial) iside a material (or grou of materials i itimate cotact) is ivariat as a fuctio of ositio de F de dy F de dz F 0 A o-zero ( built-i ) electric field is established i ouiformly doed semicoductors uder euilibrium coditios Ɛ (1/ )(de C /) (1/ )(de i /) (1/ )(de V /) bad bedig, as for a eteral field; here the field is built-i L.Laceri - Comlemeti di Fisica - Lecture 24 9 Eistei relatio - 1 Relatio betwee electro cocetratio, door cocetratio, Fermi level E F ad itrisic Fermi level E i i the uasi-eutrality aroimatio (local eutrality caot be eact, otherwise there would be o built-i field): i e E F E i At euilibrium, E F is costat bad bedig : i # E F E i kt l N & D % ( de i $ ' Iduced ( built-i ) electric field: ε dv 1 de i kt N D 1 ( ) kt N D kt N D dn D dn D E C E F E i E V L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 5

6 Comlemeti di Fisica - Lecture Eistei relatio - 2 At euilibrium the total electric curret (diffusio + drift) must be zero; usig for the electric field ε the eressio just derived oe obtais: J, 0 µ ε + D d N D dn N D ( )µ ε + D D $ N D ( )µ kt 1 dn D ( ) ' % & ( ) + D dn D Eistei relatio for electros (similar for holes): D µ kt " D kt % $ # µ ' & E C E F E i E V L.Laceri - Comlemeti di Fisica - Lecture No-uiform doig: a eamle Door cocetratio: bad bedig : Built-i electric field: Ɛ (1/ )(de i /) (1/ )(d(e F -E i )/) - dψ/ Euilibrium: E F costat Electric otetial V() Ψ() (1/ )(E F - E i ()) L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 6

7 Comlemeti di Fisica - Lecture Eamle: abrut juctio L.Laceri - Comlemeti di Fisica - Lecture juctio At thermal ad diffusive euilibrium, o eteral field: Net curret is zero, both for electros ad holes drift diffusio electros diffusio drift holes -tye -tye Built-i electric field L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 7

8 Comlemeti di Fisica - Lecture Eistei relatio umerical eamles I the o-degeerate limit: D µ kt D µ kt Tyical sizes of mobility ad diffusivity: T 300K µ 1000 cm 2 kt / Vs V D 26 cm 2 / s L.Laceri - Comlemeti di Fisica - Lecture Curret desity euatios Whe a eteral electric field Ɛ is reset i additio to the cocetratio gradiet: o euilibrium Both drift ad diffusio currets will flow The total curret desity is differet from zero i this case For electros ad holes: J J + J J J J J, drift + J, drift, diff Ɛ + J, diff µ drift + D µ Ɛ D d diffusio d L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 8

9 Comlemeti di Fisica - Lecture High field effects Drift velocity saturatio Avalache rocesses Drift velocity saturatio Silico: electros ad holes drift velocity - icreases liearly at small fields - saturates (~ thermal velocity) at high fields GaAs: electros ad holes drift velocity - Peculiar behavior - see et slide for a elaatio L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 9

10 Comlemeti di Fisica - Lecture Two-valley semicoductors GaAs: Two-valley model of E-k bad diagram: Differet effective masses Differet mobilities µ σ µ v τ c m ( µ 11 + µ 22) µ ( µ + µ ) ( + ) 1 µ E L.Laceri - Comlemeti di Fisica - Lecture To start a avalache: eough kietic eergy to create a e-h air Avalache rocesses Order of magitude estimate from eergy ad mometum coservatio (rocess 1 2): 1 2 m 1v 2 s E g m 2 1v f m 1 v s 3m 1 v f Mea free ath E k,mi > E g E m v 2 1 s 1.5 E g G A 1 ( α J + α J ) Bad bedig due to eteral field: Ɛ (1/)dE C / L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 10

11 Comlemeti di Fisica - Lecture Ioizatio rates ad geeratio rate α, α ioizatio rates (cm -1 ) e h geeratio rate (cm -3 s -1 ) G A 1 ( α J + α J ) I Silico: E E g (el.) E E g (h.) Ɛ ~ V/cm For α ~ 10 4 cm L.Laceri - Comlemeti di Fisica - Lecture Lecture 24 - eercises Eercise 1: Fid the electro ad hole cocetratios, mobilities ad resistivities of silico samles at 300K, for each of the followig imurity cocetratios: (a) boro atoms/cm 3 ; (b) boro atoms/cm 3 together with arseic atoms/cm 3 ; ad (c) boro atoms/cm 3, together with arseic atoms/cm 3, ad gallium atoms/cm 3. Eercise 2: For a semicoductor with a costat mobility ratio b µ µ > 1 ideedet of imurity cocetratio, fid the maimum resistivity ρ m i terms of the itrisic resistivity ρ i ad of the mobility ratio. Eercise 3: A semicoductor is doed with N D (N D >> i ) ad has a resistace R 1. The same semicoductor is the doed with a ukow amout of accetors NA (N A >>N D ), yieldig a resistace of 0.5R 1. Fid N A i terms of N D if the ratio of diffusivities for electros ad holes is D /D L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 11

12 Comlemeti di Fisica - Lecture Back-u slides Graded juctio N N D N A N: et fied charge 0, 0 : majority carrier cocetratios et mobile charge et total charge otetial Ψ El.field Ɛ electro eergy-bad diagram L.Laceri - Comlemeti di Fisica - Lecture L.Laceri - Comlemeti di Fisica 12