Chapter 5 Carrier transport phenomena

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1 Chater 5 Carrier trasort heomea W.K. Che lectrohysics, NCTU Trasort The et flow of electros a holes i material is calle trasort Two basic trasort mechaisms Drift: movemet of charge ue to electric fiels Diffusio: the flow of charges ue to esity graiet We imlicitly assume the thermal equilibrium urig the carrier trasort is ot substatially isturbe W.K. Che lectrohysics, NCTU

2 Outlie Carrier rift Carrier iffusio Grae imurity istributio The Hall effect W.K. Che lectrohysics, NCTU 3 5. Carrier rift Drift: the et movemet of charge ue to electric fiels is calle rift Drift curret: The et rift of charge gives rise to a rift curret l A l N: total umber of flow charge : volume esity of flow charge A: cross-sectioal area υ: average rift velocity l: travelig legth of carrier er Δt N Al A( υδt # Flu Φ Aυ ( Δt Δt Δt sec Φ # Flu esity φ υ ( A sec A Drift curret esity qφ qυ ( Amere W.K. Che lectrohysics, NCTU 4

F r et mobility The average rift velocity for low electric fiels is irectly roortioal to the electric fiel, similar to the termial velocity case i Fuametal Physics Hole: * F m a ( + e r υ μ r :

3 F r et mobility The average rift velocity for low electric fiels is irectly roortioal to the electric fiel, similar to the termial velocity case i Fuametal Physics Hole: * F m a ( + e r υ μ r : electric fiel υ : average rift velocity for holes μ : hole mobility, roortioality factor crystal r F et * m a The mobility escribe how well a article will move ue to a electric fiel Drift curret ue to holes ( + e υ ( + e μ, rf W.K. Che lectrohysics, NCTU 5 qφ qυ r υ μ r A ( F ( e, rf qυ ( e υ ( e ( μ Flu φ Flu φ F (+ e, rf, rf Drift curret ue to electros, rf eμ Total rift curret + ( eμ + eμ, rf, rf W.K. Che lectrohysics, NCTU 6

4 amle 5. rift curret esity 6 3 GaAs samle at 300K, N a 0, N Assume comlete ioizatio Calulate the rift curret esity if alie electric fiel V/ Solutio: comlete ioizatio N 6 6 i ( Calulate the rift curret esity if alie electric fiel V/ -3 ft, rf +, rf ( eμ + eμ ft (.6 9 (8500( 6 ( 36A/ W.K. Che lectrohysics, NCTU Mobility effect F F * υ m a m ( + e υ t * e * m t t Uer thermal equilibrium Assume the et rift velocity is a small erturbatio o the raom thermal velocity, so the time betwee collisio will ot be altere areciably eτ mea eak velocity, * c υ eak m τ c : mea time betwee collisios W.K. Che lectrohysics, NCTU 8

5 The average rift velocity is oe half the eak value average eτ velocity * c υ m Due to the statistic ature, the factor of ½ oes ot aear i a more accurate moel average velocity υ μ υ μ eτ c υ μ * m eτ m eτ m The less the collisios, the loger the mea collisio time a the higher the mobility c * c *, W.K. Che lectrohysics, NCTU 9 Scatterig (collisio mechaisms Two major scatterig mechaisms Phoo (lattice scatterig A erfect erioic otetial i a soli allows electros to move uimee, or with o scatterig, through the crystal The thermal vibratios of lattice atoms cause a isrutio i the erfect erioic otetial, resultig the iteractios betwee the electros or holes a the vibratig lattice atoms μ L lattice oize imurity scatterig 3/ T The imurites i semicouctor at higher temeratures. The coulomb iteractios betwee the electros or holes a the ioize imurities rouce scatterig or collisios. total ioize imurity T 3/ μ imurity N N ( + N + Na W.K. Che lectrohysics, NCTU

Mobilitis versus temerarure serts show the temerature eeece

However, the mobilites o icrease as the temerature icreases

6 Mobilitis versus temerarure serts show the temerature eeece for almost itrisic silico The iserts show that the arameter is ot equal to 3/, but is., as the first-orer scatterig theory reicte. However, the mobilites o icrease as the temerature icreases W.K. Che lectrohysics, NCTU Mobilties versus imurity cocetratios at 300K Ge Si GaAs W.K. Che lectrohysics, NCTU

Net mobility The robability of a scatterig eve i the ifferetial time t is the sum of iiviual evets t t t + τ τ τ υ eτ c υ eτ c Qμ, μ μ τ * * m m The et mobility ue to the ioize a lattice scatterig

7 Net mobility The robability of a scatterig eve i the ifferetial time t is the sum of iiviual evets t t t + τ τ τ υ eτ c υ eτ c Qμ, μ μ τ * * m m The et mobility ue to the ioize a lattice scatterig rocesses L + μ μ μ L High effective mass of carrier results i low mobility The mobility will icrease with the icreasig collisio time W.K. Che lectrohysics, NCTU Couctivity e( μ + eμ σ V R, R ρ L L A σa A V / R A L A L σa σ σ σ ( eμ ρ + eμ σ e( μ + eμ The couctivity a resistivity of a etrisic semicouctor are a fuctio rimarily of the majority carrier arameters, such carrier cocetratios a mobilities W.K. Che lectrohysics, NCTU 4

Resistivity versus imurity cocetratio at 300K

essetially costat, However, the mobility ecreases with icreasig

8 Resistivity versus imurity cocetratio at 300K Si Ge, GaAs a GaP W.K. Che lectrohysics, NCTU 5 σ ( e μ + eμ i the mitemerature rage (etrisic rage We have comlete ioizatio, the electro cocetratio remais essetially costat, However, the mobility ecreases with icreasig temerature At higher temeratures The itrisic carrier cocetratio begis to omiate the electro cocetratio as well as the couctivity W.K. Che lectrohysics, NCTU 6 i

9 amle 5. mobility Comesate - tye Si at 300K, σ 6 ( Ω -, Na 6 Determie the oor cocetratio a electro mobility Solutio: comestae semicouctor a comlete ioizatio at 300K ( N σ eμ eμ ( N 6 (.6 The N 5 9 a N μ ( N 3 7 /V -s a -3 7 Use the left figure with trial a error For eamle if we choose N N μ + N a 3 ( i. e., N 7 σ 8.6 ( Ω W.K. Che lectrohysics, NCTU 7 f we choose N The N μ + N a /V -s -3 ( i. e., N 5 7 σ 0.8 ( Ω Further tial a error yiels N N N μ + N a 400 σ 6 ( Ω /V -s -3 (agree with give value W.K. Che lectrohysics, NCTU 8

10 5..4 Velocity saturatio 3 mυ th kt at T 300 K 3 mυ th ( eV υ th 9 ( eV ( ( m (9. m/s υ th 7 /s W.K. Che lectrohysics, NCTU 9 Figure 5.8 For Si At low electric fiels, there is liear variatio of velocity with electric fiel At high electric fiels, the velocity saturate at aroimately 7 /s W.K. Che lectrohysics, NCTU 0

11 GaAs: For GaAs m 0.067m Si : m. 08m * o Due to low effective mass, the low-electric fiel electro velocity i GaAs is much larger tha i Si. At high electric fiels, egative ifferetial mobility occurs ue to the scatterig of electros ito uer valley. Because of larger effective mass i the uer valley (0.55 mo vs mo, the itervalley trasfer mechaism results i ecreasig average rift velocity of electros with electric fiel. W.K. Che lectrohysics, NCTU * o 5. Carrier iffusio ( ( l υ τ th c ( o o l o o + l o F υ l th W.K. Che lectrohysics, NCTU

12 Durig electro travel betwee collisios a mea free time, Oe half of electros at segmet ( will move to the right a cross the o lae ito segmet ( Oe half of electros at segmet ( will move to the left a cross the o lae ito (segmet ( l ( ( Net rate of electro flow to the right (electro flu F ( o υ th υ th ( υth F ( υth ( l et electro flu l υ F thl o l o l υ τ th o + l c W.K. Che lectrohysics, NCTU 3 lectro iffusio curret ( e F + eυ l th, if, if ed ed Hole iffusio curret electro iffusio coefficiet D υthl l υ τ c hole iffusio coefficet l υ D l thτ υth th c W.K. Che lectrohysics, NCTU 4

13 amle tye GaAs at 300K, The electro cocetratio varies liearly from istace of 0. Calculate the iffusio curret if Solutio: 8 iffusio coefficiet to 7 D over a /s, if ed ed Δ Δ (.6 ( A/ W.K. Che lectrohysics, NCTU Total curret esity eμ + eμ +, if + ed ed eμ + eμ +, if + ed ed W.K. Che lectrohysics, NCTU 6

14 5.3 Grae imurity istributio ouiform oe semicouctor, there will be a iffusio of majority electros from the regio of high cocetratio to the regio of low cocetratio. The flow of electros leave behi ositively charge oor ios. The searatio of ositive a egative charges iuces a electric fiel The electric fiel is efie as The electric otetial is relate to the electric otetial eergy by the charge ( -e φ c ( e φ e φ Fi ( F Fi e W.K. Che lectrohysics, NCTU 7 o i f e kt fi N ( f fi N ( kt l i kt N ( N ( fi The iuce electric fiel ue to the ouiform oig kt e N ( N ( W.K. Che lectrohysics, NCTU 8

15 amle 5.5 N ( 6 9 ( -3 (0 μm Solutio: kt 9 N ( (0.059( 6 ( ( e N 9 At 0, we fi ( V/ W.K. Che lectrohysics, NCTU The istei relatio thermal equilibrium, the iiviual electro a hole curret must be zero 0 eμ + ed 0 eμ + ed N ( 0 kt N ( eμ ed e N ( + N ( Q( N ( D μ kt e D μ D μ kt e W.K. Che lectrohysics, NCTU 30

The hole curret must also be zero istei relatio D μ D μ kt e 0.056 at T 300 K μ 40 at T 300K D W.K. Che lectrohysics, NCTU 3 5.

16 The hole curret must also be zero istei relatio D μ D μ kt e at T 300 K μ 40 at T 300K D W.K. Che lectrohysics, NCTU The Hall effect The Hall effect is use to istiguish whether a semicouctor is -tye or - tye To measure the majority carrier cocetratio a Majority carrier mobility q y F B qυ q B F q[ + υ B] 0 H qυ B V + W H H z The iuce electric fiel i the y-irectio is calle the Hall fiel The iuce electric fiel rouce a voltage i the y-irectio is calle the Hall Voltage V H υ WB :rift velocity z υ W.K. Che lectrohysics, NCTU 3

17 Semicouctor tye & cocetratio For -tye semicouctor, the rift velocity is relate to material arameters υ e ( e( W V H Bz e B ev z H V H Bz e B ev z H W.K. Che lectrohysics, NCTU 33 mobility Oce the majority carrier cocetratio is etermie, we ca calculate the low-fiel majority carrier mobility Q eμ W V eμ L μ L ev W μ L ev W W.K. Che lectrohysics, NCTU 34

18 amle 5.7 carrier cocetratio & mobility L 0.,.0 ma, V W,.5 V, B z gauss 5 tesla, a V H 6.5 mv Solutio: B ev z H 3 ( (5 (.6 ( ( m μ L ev W μ ( ( ( (5 (.5( ( 0. m /V -s μ 00 /V -s W.K. Che lectrohysics, NCTU 35 Figure 5.4 Figure for Problem 5. W.K. Che lectrohysics, NCTU 36

MODULE 1.2 CARRIER TRANSPORT PHENOMENA

MODULE 1.2 CARRIER TRANSPORT PHENOMENA Carrier Trasort Pheoeo Carrier drift: obility, coductivity ad velocity saturatio Carrier Diffusio: diffusio curret desity, total curret desity The Eistei relatio