Low field mobility in Si and GaAs

Transcription

1 EE30 - Solid State Electronics Low field mobility in Si and GaAs In doed samles, at low T, ionized imurity scattering dominates: τ( E) m N D πe 4 ln( + γ ) γ γ E 3 τ o ( E kt) 3 where τ o ( kt) 3 m , and. N D πe 4 ln( + γ ) γ _ 8m EL + γ D h γ τ o is aroximately constant. Plug in Ê 3kT (maximum of integrand in Eτ( E) ). Define: The we get for the mobility: μ B γ B e Γ( 4) m m Γ( 5 ) N D πe 4 ln( + γ B ) γ B ( kt) 3 Notice that this increases with temerature ~ ( kt) 3. Faster electrons get deflected less. Acoustic honons: (Moderate T, mainly imortant in Si hω LO ( Si) 63meV) πh _4 ρv s τ o D A ( m ) 3 ( kt) 3 μ AP πh _4 ρv s e ( m ) 5 ( kt) 3 D A Γ( ) Γ( 5 ) - 3 -

2 EE30 - Solid State Electronics AP mobility decreases with increasing temerature as (kt) -3/ - increased honon oulation Multile scattering mechanisms (read Lundstrom 4.3, 4.7, 3.7) relaxation - time aroximation: f --- t coll f( E) fe ( ) fe ( ) τ ( E) τ ( E) τ eff ( E) recall μ e e τ m eff ( E) Eτ eff( E) m E As we have seen, for most scattering mechanisms: τ i ( E) τ oi ( E kt) s i, and if s s s both mechanisms have same E - deendence, then -- μ m e Γ( 5 ) Γ( 5 ) Γ( s + 5 ) Γ( s + 5 ) τ o τ o so Mathiessen s rule This is only true if both mechanisms have the same E - deendence. This is rarely the case, but Mathiessen s rule is often used anyway, since it s hard to do otherwise. Only matters when several rate s are comarable. Often one rocess dominates. Exect for Si Im: log (μ) 3 T N ADP: T 3 N N 3 log T - 4 -

3 EE30 - Solid State Electronics logμ GaAs im POP ( kt) N N N log T High field transort High field effects: qualitative discussion velocity saturation: tyical in Si v d μe. For low fields, μ μ 0, indeendent of E. At high fields, the drift energy becomes large enough that equivalent intervalley otical honon-scatt turns on. This creates a strong energy loss for electrons with velocity. This leads to a rolloff of the drift velocity, and eventually, full saturation. v d v d μ 0 E v dsat velocity overshoot (in Si) For time scales shorter than the equivalent intervally scattering time, τ ot, a transient drift velocity, v d > v dsat can occur. Equivalently, in devices that are small enough that electrons can traverse the device structure in such a short time, the average roagation velocity can E - 5 -

4 EE30 - Solid State Electronics v dsat also exceed. Velocity Overshoot (GaAs). In GaAs, there is no intervalley rocess for E < 0.9eV n L, m L, μ L Δ n Γ, m Γ, μ Γ For low energies, POP scattering dominates. Recall that the rate for POP scattering, /τ, is nearly indeendent of electron energy. It is also fairly weak. So the low field mobility in GaAs is rather high, about 8000 cm /V-s. Once the electron energy reaches 0.9 ev, the intervalley scattering rocess (Γ L) turns on. The mobility in the L valley is much lower, about 500 cm /V-s. Also, for electrons in the L-valley, intervalley scattering to equivalent L valleys can occur, as well as scattering back to the Γ-valley. Hence, the velocity for electrons in the L-valley is saturated, much as in silicon. V v μ Γ E intervalley transfer strong low mobility in L Balance Equations (read Lundstrom Ch. 5) Our aroach here is to go back to the full BTE, which is a balance equation for the full distribution f(). Using a rocedure to be defined, we will take moments of the BTE. In this way, we sill derive balance equations for ensemble average quantities, which will be couled artial differential equations (PDE s). Recall the full BTE (assume we are considering electrons, q e ) - 6 -

5 EE30 - Solid State Electronics Consider some function of momentum φ( ). The ensemble average of φ( ) is given by: n φ ( rt, ) -- φ( )frt (,, ) V To get a balance equation for, take the φ( ) moment of BTE. That is, take n φ. Exanding this out exlicitly: -- φ( ) --- f -- V + φ( )v f t V r -- φ( )qe f V n φ Fφ t F φ -- φ( )vf V flux by arts -- φ( ) f V --- t qe φf f φ V f 0 as ± Grt (, ) qe V -- f φ coll Consider the collision term: We will write this as: ---- τ φ 0 R φ ---- [ n φ ( rt, ) n φ( r, t) ] is a generalized ensemble relaxation rate. A straightforward maniulation shows that if we define the generalized relaxation rate as: τ φ φ( ' ) τ φ ( ) S' (, ) φ( ) ' - 7 -

6 EE30 - Solid State Electronics then the ensemble relaxation rate is: -- V f ( )φ( ) τ φ ( ) τ 0 φ [ n φ ( r, t) n φ( r, t) ] We will make a generalized relaxation time aroximation (RTA) and take as either a constant, or a simle function of E. This gives us the General Balance Equation of, which is in the desired form of a PDE: ---- τ φ n φ Density Balance eqn n n φ φ( ) J F φ -- vf V nv n d ( q) (electrons) G φ 0 since φ( ) 0 R φ collisions conserve total n τ φ We thus obtain the current continuity equation: Momentum Balance eqn: φ( ) z (just one comonent of the momentum) momentum density F φ -- v momentum flux ( kinetic energy) V z f W z G φ qe V -- f z qne z momentum generation - 8 -

7 EE30 - Solid State Electronics R φ [ P z 0] τ m P z τ m P z W z qne t z ---- P z generalizing to the full 3 comonents of momentum: τ n Note: Assuming isotroic arabolic band: W : vector W ( W) ij j x i The momentum balance equation then is: The kinetic energy tensor is given by: The kinetic energy density is given by: tr( W) W ij W ii i W ij v V i j f i -- V energy density --m v i f nw W avg ke er carrier i nm v i - 9 -

8 EE30 - Solid State Electronics Write the velocity as two arts: with c 0. Then: tr( W) --nm v d + --nm c W drift + W thermal We define T eff, the effective temerature via: T eff has its usual meaning if f(c) is a Maxwell-Boltzmann distribution. Rewrite W: W ij K ij + --nkt ij This gives a generalized definition for drift energy and temerature tensors: K ij n --m v di v dj such that: --nkt ij --nm c i c j Aroximations We now make a set of reasonable aroximations to make this equation tractable: () --nkt» K Thermal motion dominates the energy. This assumtion will have to be re-examined for devices with ballistic effects () T Thermal energy is equally artitioned among all 3 degrees of freedom. T e

9 EE30 - Solid State Electronics Under this aroximation, we have the simlification that T T e (3) Current varies slowly comared to the time scale: (quasi - steady state) τ m With these aroximations, the momentum balance eqation becomes: J n e ne e ( nkt e ) m m τ m τ m Defining the transort arameters: e kt μ n e k D m τ m n μ e n S n nμ n - e we get the drift-diffusion equation: (Soret coeff) To cature nonthermal henomena need at least one more moment of the BTE. Energy Balance Equation φ( ) E ( ) n φ -- 3 E ( )f V W tr( W) --nm v d + --nkt e G φ ee V -- E( ) f ee -- vf E J V n velocity as we had before this is called the energy flux Putting these terms together, we get the energy balance equation: - 3 -

10 EE30 - Solid State Electronics Let us examine the energy flux more closely: F w m v vf V First slit the velocity into drift and random comonents as before: nm v v F w nm [ v v d + v c ] Wv d ( v d + v d c+ c )c Now define a quantity we will call heat flux : We can now exress the energy flux as: F w Wv d + v d nkt + Q We now make a further set of aroximations: () Q κ r T e (κ : thermal conductivity) This is somewhat ad hoc, but lausible. () W 3 as before --nkt e The resulting exression for the energy flux becomes: Wiedemann-Franz law relates electrical to thermal conductivity: κ ( s + 5 ) ( k e) σt e Since J n env d, and σ neμ, we have: - 3 -

11 EE30 - Solid State Electronics 5 F w -- kt e J n s 5 kμn T We finally obtain a simle form for the energy balance equation: Energy balance treatment of velocity saturation (read Lundstrom 7.) At high electric fields, a non-equilibrium condition is roduced in which the electron temerature and lattice temerature are not necessarily equal. We must be careful to kee track of each searately, as T L and T e. Consider the ADP momentum relaxation time: In the above exression, T reresents T L since it relates to the honon oulation. However, in calculating the weighted average electron distribution. When we write: --, we average over f(e), which is the τ and _4 πρv s h τ ( m kt) 3 D A in order to roerly handle the high field case, we must write: τ m ( E) _4 πρv s h ( m kt L ) 3 D A T L T e () E () kt e τ 0 ( E kt e ) Comaring () and (), we see that when, T e T L

12 EE30 - Solid State Electronics where τ m is the low-field relaxation time. Finally, the ADP hot carrier mobility is seen to be given by: (3) where μ 0 is the lowfield mobility. Examining (3) we see that as the electron temerature increases (which is what occurs as the field increases), the mobility decreases. This leads to velocity saturation. Thus we learn that velocity saturation can indeed come from acoustic honon scattering. As it turns out, otic honon scattering is actually not necessary. Any time the mobility decreases with increasing electron temerature, we can get velocity saturation. Now we need to figure out how T e deends on the electric field. We will us an energy balance model in which two arameters describe the electron energy distribution: v d, and T e. This essentially assumes that the distribution is described by the drifted Maxwellian. We have two relaxation times τ m (momentum relaxation time) and τ E (energy relaxation time). We will simultaneously solve the couled momentum balance and energy balance equations. From momentum balance (drift-diffusion) equation (assume satially uniform field directed in the z-direction, and steady state): J n qnv dz With the same assumtions, the energy balance equation becomes: Plugging in for J: J n 3 E --nk( T e T L ) ---- τ E

13 EE30 - Solid State Electronics q ne m τ m 3 --nk( T e T L ) ---- τ E T e q E T L 3m kt L τ m τ E qμ 3kT L ---- τ E The energy relaxation is dominated by equivalent intervalley scattering in silicon. Due to the discontinuous form of the scattering rate, the calculation is non-trivial, but a good aroximation for the ensemble relaxation rate is given by: For further detail see reference [7.] in Lundstrom. With the hot carrier mobility determined by acoustic honon scattering: T μ μ L T e T e μ 0 T L T e q E T L 3 kt L C kt L T L T e Thus: qμ E C

14 EE30 - Solid State Electronics Going back to the hot-carrier mobility: T μ μ L T e So, we find for the drift velocity: v d μe μ 0 E ( E E cr ) Exerimentally, it is found that the high field mobility can be fit by a slightly generalized form of this known as the Thornber equation: v d v sat μ 0 E cr β..4 ε

15 EE30 - Solid State Electronics Monte-Carlo simulation (read Lundstrom, Chater 6) - solves full Boltzmann - concetually simle - comutationally very costly The idea is to follow individual article trajectories semi-classically. - calculate classical free flight between collisions (using bandstructure information). - a suitably weighted random # generator simulates the random quantum mechanical scattering rocesses. It is used to determine free flight intervals, and the results of scattering events. Free flight - rt () r( 0) + vt' ( ) dt' You can ut in as much of the bandstructure details into the effective mass tensor as your comuter can stand. Scattering: t 0 Here again, you ut in as many scattering rocesses as you can manage. Pick a Γ 0, indeendent of, Γ 0 > max[ Γ( ) ] This is the candidate scattering rate. The robability that a carrier ossibly scatters at time t is given by: Pt () Γ 0 e Γ 0t. Monte Carlo algorithm: () Using random # generator, ick a scattering time according to P(t) distribution. () Udate article r, using free flight algorithm (bandstructure). (3) Using random # generator, ick a scattering rocess weighted by Γ i ( ). Include no

16 EE30 - Solid State Electronics scattering at rate Examle weighting factors: Γ i ( ) P i Γ 0 ADP.7% equiv intervalley ot. abs.9% equive intervalley ot. emiss. 0.3% Imurity.% no scatt 83.9% (4) Udate magnitude & direction of. Use S' (, ) robabilty (RNG) (5) Go to () Self consistent Monte Carlo Take into account the change in electric fields as the carriers move. - discretize sace into a grid. - oulate device with N electrons (more the better!) - to kee N reasonable, assign charge to each electron Q e N act actual # N N act electrons in device region

17 EE30 - Solid State Electronics initial condition Aly bias field move all articles by MC method for time Δt solve Poissons eqn in new config. Get fields no done? yes Drawbacks of Monte Carlo: () Need large N for good statistics, which is exensive in comuter time () statistical uncertainty reduces only as (3) Exonentials! N many device characteristics deend on exonential behavior in the tail of the distribution - i.e. MOSFET turn-off, MOSFET hot carrier degradation. This demands very large N