Conductance from Transmission Probability

Transcription

1 Conductance rom Transmission Probability Kelly Ceung Department o Pysics & Astronomy University o Britis Columbia Vancouver, BC. Canada, V6T1Z1 (Dated: November 5, 005). ntroduction For large conductors, conductance ollows Om s Law G σw/l were G conductance, σ conductivity, wic is independent o te sample dimensions, W widt o te conductor, and L lengt o te conductor. Tereore, tis law predicts tat wen L goes to 0, G goes to ininity. Unortunately, G -1 experimentally approaces a minimum value o G C -1 wic will be explained to be contact resistance. A new conductance ormula: G (e /)MT were M is te number o modes, and T is te probability tat an electron transmits troug te conductor, is sown to produce te correct conductance or conductors o bot large and small scales. n tis paper, te new conductance ormula will be derived along explaining its ramiications to electron energy distribution, voltage drop, and eat dissipation across conductors.. Ballistic Conductor A. Contact Resistance To discover wat is contact resistance, we will examine te setup sown in igure 1. Tere are two contacts joined by a conductor o lengt L and widt W. At te contacts, tere is an ininite number o transverse modes tat carry te current, wile in te conductor, tere are only a ew modes. Te redistribution o current at te interace is wat leads to a minimum resistance called contact resistance G C -1. Figure 1: Setup were conductor is placed between two contacts 1

2 Figure : Tere is an ininite number o modes in te contacts, but only a ew modes in te conductor B. Derivation o Contact Resistance We will now derive te value o G C making two assumptions: ballistic conductor and relectionless contacts. For a ballistic conductor, te probability o transmission or electrons troug te conductor is unity. te contacts are relectionless, tere are no relections o te electrons entering te contacts rom te conductor, but tere is still relection o te electrons entering te conductor rom te contacts. Relectionless contacts are sown by numerical calculations to be a good approximation as long as te electrons are not too close to te bottom o te energy band. We now look at current wen we ave apply a bias across te contacts, µ 1 on te let contact, contact 1, and µ on te rigt contact, contact. Te +k states in te conductor will ave energy µ 1 and te -k states in te conductor will ave energy µ. Te reason or tis is i we apply te same energy µ 1 on bot contacts, obviously te +k and k states will ave te same potential. Now, let us cange te potential on contact to µ. Te k states rom te second contact will ave te new potential, but te energy o te +k states will not be aected since tere is no causal relationsip between te +k and k states due to relectionless contacts. As well, te current is carried only by te occupied states between µ 1 and µ since te +k and k states below te lower energy will cancel. Figure 3: Only modes between µ1 and µ produce current Notice tat or eac transverse mode, labeled by N, tere is a dispersion relation E(N,k) wit a

3 cuto ε E(N,k0) below wic te mode cannot propagate, like waves in a waveguide. Hence, tere are M propagating modes given by M ( E) θ ( E εn ) N were θ is te step unction. Examining current or a particular mode, env e L e L e k L π ε 1 dk dk dk ( E) ( E) ( E) were e electron carge, n electrons per lengt, v electron velocity, and (E) is electron distribution. Te integral is rom ε to ininity since ε is te lowest energy or tat mode. Now, or more tan one mode, current is simply: e ( E) M ( E) M is constant over te bias µ 1 to µ, e ( µ 1 M e G V µ C ) Tereore, e G C M C. Step Response and Voltage Drop We know M, assuming periodic boundary conditions, because te propagating k must be between k and k, were k is te Fermi momentum. Tereore, or widt W, te separation between k values is π/w, and tereore, 3

4 M M M k π / W kw π W λ / were λ Fermi wavelengt. By decreasing te widt o te conductor, te number o modes decreases, and we see te contact resistance drops in steps o e / as seen in igure 4. For metals, λ is small; tereore, are many modes so wen te widt canges, tere is a negligible cange in current. On te oter and, λ can be quite large in semiconductors so we can see experimentally te step beaviour. Figure 4: Conductance as a unction o Gate Voltage (wic controls te widt o te conductor) Tere is no voltage drop across te conductor, but tere is a drop o (µ 1 -µ )/e across te contacts. For te +k states, te +Fermi level, +, drops at te rigt contact wile or te k states, te -Fermi level, -, canges at te let contact. Let us deine voltage to be te average Fermi level or te +k and k states. 4

5 Figure 5: Voltage drops at te interace between te conductor and te contact. Landauer Formula A. Proo o Landauer Formula We now consider conductors were te transmission probability, T, is not unity. Let us guess tat G e / MT, te Landauer ormula, and sow tat we arrive at te correct conductance relationsips or large and small conductors. Notice tat tis ormula matces te one derived or te ballistic conductor (T 1) and makes sense or T 0; conductance is zero wen tere is no transmission. Proo: e G e G e G π MT kw T π πnsw m T π m πnsw G e TNs π e vwtns G π were k πns, were n s electron density, N s m / π, v π n s / m e vwns G π L+ e vwns G π L+ D G ewns L+ D G Wσ L+ 5

6 Using te Einstein relation by writing current density as relating drit velocity to electric ield ten comparing section, we will prove T /L+ were is a constant. Tereore, J en s vd, were v d drit velocity, and J σe, we ind σ e N s D. n te ollowing G G G L+ WσD L + WσD 1 G s + G c WσD 1 were G s -1 is te resistance rom Om s law and G c -1 is te contact resistance, and we arrive at te correct ormula or conductance. Figure 6: Setup similar to tat or te ballistic conductor except tere is a transmission probability B. Derivation o T Let us now consider two conductors in series, eac wit a scatterer. Te probability o transmission troug te irst scatterer is T 1 and te probability o transmission troug te second scatterer is T (and relection probabilities R 1 and R ). Te probability o transmission troug bot conductors T 1 is not T 1 T. tis was te case, or a cain o scatterers, te probability o passing troug all te scatterers would decrease exponentially and we would not arrive at Om s law. Tere are an ininite ways o transmitting troug te two conductors: directly T 1 T, relecting twice T 1 R1R T, relecting our times, T 1 R 1 R R 1 R T, etc as sown in igure 7. Summing tese probabilities, T 1 T 1 T /(1-R 1 R ) 6

7 Figure 7: Transmission troug two conductors, eac wit a scatterer Te expression (1-T 1 )/T 1 (1-T 1 /)/T 1 +1-T /T as an additive property. Tereore, n general or N scatterers, 1-T(N)/T(N) N(1-T)/T wen we take te transmission probabilities or eac scatterer to be te same. Tereore, T ( N) T N(1 T ) + T T T ( N) νl(1 T ) + T T ( N) L+ were NυL were υ is te linear density o te scatters. C. Energy Distribution and Voltage Drop We now consider te energy distribution o te electrons ar away and near te scatterer. On te let o te scatterer, te +k states are ave te same energy as te let contact µ 1. Similarly, on te rigt o te scatterer, te k states ave energy µ. On te rigt just ater te scatterer, +k states rom 0 to µ 1 are illed proportional to T. n addition, +k states are illed rom te relection o te k states rom 0 to µ proportional to 1-T. Similarly or te k state electrons near te scatterer on te let side but reversing te probabilities. Far rom te scatterer on te rigt, te electrons in te +k states redistribute teir energy to occupy te lower energy and reac a new igest energy level o F wile te k ar let o te scatterer redistribute to F as sown in igure 8. Notice tat F µ +(1-T)( µ 1 -µ ) and F µ +T(µ 1 -µ ). 7

8 Figure 8: Energy distribution o electrons ar and near scatterer Tereore, one gets te ollowing Fermi levels or te +k and k states: + Let: ( E ) θ ( µ 1 E ) ( E θ µ E + T θ µ 1 E θ µ E Near let: ) ( ) (1 ){ ( ) ( )} ( E θ F E Far let: ) ( ' ) ( E θ µ E Rigt: ) ( ) ( E θ µ E + T θ µ 1 E θ µ E + Near rigt: ) ( ) { ( ) ( )} ( E θ F E + Far rigt: ) ( '' ) te energy o te +k electrons is µ1 beore te scatterer and µ +T(µ 1 -µ ) ater te scatterer, te potential dierence across te scatterer is just te dierence (1-T) (µ 1 -µ )/e. We get te same result i we consider te k states. Tereore, te resistance is at te scatterer. Notice tat te remaining potential drop wen compared to te original bias o µ 1 -µ is T(µ 1 -µ ) wic is exactly te voltage drop across te contact. Figure 9: Voltage Drop across scatterer is (1-T)(µ 1-µ ) and across contact is T(µ 1 -µ ) D. Heat Dissipation Scatterers are assumed to be rigid and ave no internal degrees o reedom; tereore, tey cannot 8

9 dissipate eat. Te dissipation o eat / scatterer. Rewriting te equation or eat dissipation, G S is rom te evolution o energy distribution ater te P P D D d U dz du e dz were P D eat dissipation, energy current i E) (, U and net current. 1 U Ei( E) e, average energy o te current U U U e U Ei( E) i( E) E(eM / )( (em / )( + + ) ) Tereore, ' µ1 F+ µ 1 + µ '' + µ F U on te ar let U near te scatterer U on te ar rigt Since tere is eat dissipation as long as te electrons ave not yet reaced equilibrium ar rom te scatterer because du/dz is not zero, te dissipation is not only at te scatterer. 9

10 Figure 10: Heat dissipation (were te average energy o te current canges) is not at te scatterer V. Reerences Datta, Supriyo. Electronic Transport in Mesoscopic Systems. Cambridge, New York, (003). 10