Scharfetter Gummel Schemes for Non-Boltzmann Statistics

Transcription

1 Scharfetter Gummel Schemes for Non-Boltzmann Statistics Patricio Farrell Weierstrass Institute (WIAS) Joint work with: T. oprucki, J. Fuhrmann, N. Rotundo, H. Doan ECMI 2016

2 Numerical methods for interesting physics Boltzmann Blakemore Fermi-Dirac e 1 e + 1 ( +1) 1R 0 E 1+e (E ) de 8 6 F( ) η

3 Numerical methods for interesting physics Boltzmann Blakemore Fermi-Dirac e 1 e + 1 ( +1) 1R 0 E 1+e (E ) de 8 6 F( ) Physics and numerics easy η

4 Numerical methods for interesting physics Boltzmann Blakemore Fermi-Dirac e 1 e + 1 ( +1) 1R 0 E 1+e (E ) de 8 6 F( ) Physics and numerics easy 4 2 Physics and numerics complex η

5 Outline semiconductor equations (arb. distribution function) and their discretisation discuss three numerical flux approximations assess quality by looking at a benchmark

6 Semiconductor Equations (van Roosbroeck) For the electrostatic potential, electron and hole densities r ("r ) =q(c + p n) 1 n t q r j n = R(n, p) p t + 1 q r j p = R(n, p)

7 Semiconductor Equations (van Roosbroeck) For the electrostatic potential, electron and hole densities r ("r ) =q(c + p n) 1 n t q r j n = R(n, p) p t + 1 q r j p = R(n, p) = change in mass flow through boundary

8 Relationships Currents Chemical potentials Diffusion and mobility

9 Relationships Currents j n = qµ n nr + qd n rn and j p = qµ p pr qd p rp Chemical potentials Diffusion and mobility

10 Relationships Currents j n = qµ n nr {z } advection + qd n rn {z } di usion and j p = qµ p pr {z } advection qd p rp {z } di usion Chemical potentials Diffusion and mobility

11 Relationships Currents j n = qµ n nr + qd n rn and j p = qµ p pr qd p rp n = N c F Chemical potentials q( 'n )+E ref E c q( 'p )+E ref E v and p = N v F k B T k B T Diffusion and mobility

12 Relationships Currents j n = qµ n nr + qd n rn and j p = qµ p pr qd p rp Chemical potentials n = N c F( n ) and p = N v F( p ) Diffusion and mobility

13 Relationships Currents j n = qµ n nr + qd n rn and j p = qµ p pr qd p rp Chemical potentials n = N c F( n ) and p = N v F( p ) D n µ n U T = n N c (F 1 ) 0 Diffusion and mobility n and N c D p µ p U T = p N v (F 1 ) 0 p N v

14 Relationships Currents j n = qµ n nr + qd n rn and j p = qµ p pr qd p rp Chemical potentials n = N c F( n ) and p = N v F( p ) D n µ n U T = g Diffusion and mobility n and N c D p µ p U T = g p N v

15 Relationships j n = qµ n nr U T g n N c Currents rn and j p = qµ p pr + U T g p N v rp Chemical potentials n = N c F( n ) and p = N v F( p ) D n µ n U T = g Diffusion and mobility n and N c D p µ p U T = g p N v

16 Relationships Currents j n = qµ n nr' n and j p = qµ p pr' p Chemical potentials n = N c F( n ) and p = N v F( p ) D n µ n U T = g Diffusion and mobility n and N c D p µ p U T = g p N v

17 Relationships Currents j n = qµ n nr' n and j p = qµ p pr' p Chemical potentials drift and diffusion quasi Fermi potential n = N c F( n ) and p = N v F( p ) D n µ n U T = g Diffusion and mobility n and N c D p µ p U T = g p N v

18 Relationships Currents j n = qµ n nr' n and j p = qµ p pr' p Chemical potentials drift and diffusion quasi Fermi potential n = N c F( n ) and p = N v F( p ) densities chemical potentials D n µ n U T = g Diffusion and mobility n and N c D p µ p U T = g p N v

19 Relationships Currents j n = qµ n nr' n and j p = qµ p pr' p Chemical potentials drift and diffusion quasi Fermi potential n = N c F( n ) and p = N v F( p ) densities chemical potentials D n µ n U T = g Diffusion and mobility n and N c D p µ p U T = g p N v nonlinear

20 Voronoi Cells

21 Voronoi Cells Flux along 1D edge x x L L

22 Finite Volume Discretisation r ("r )+q(c + p n 1 q r j n + R(n, p + 1 q r j p + R(n, p) =0

23 Finite Volume @t Z Z r ("r )dx + q 1 ndx q pdx + 1 q Z Z r j n dx + r j p dx + (C + p n)dx =0 Z Z R(n, p)dx =0 R(n, p)dx =0

24 Finite Volume @t Z Z r ("r )dx + q 1 ndx q pdx + 1 q Z Z r j n dx + r j p dx + (C + p n)dx =0 Z Z R(n, p)dx =0 R(n, p)dx =0

25 Finite Volume Discretisation Z r ("r )dx + q (C + p n p + 1 q Z Z r j n dx + R(n,p )=0 r j p dx + R(n,p )=0

26 Finite Volume Discretisation Z r ("r )dx + q (C + p n p + 1 q Z Z r j n dx + R(n,p )=0 r j p dx + R(n,p )=0 x x L L

27 Finite Volume @t n p + 1 q n ("r )ds + q (C + p n )=0 n j n ds + R(n,p )=0 n j p ds + R(n,p )=0 x x L L

28 Finite Volume Discretisation X L (x L x ) ("r )+q (C + p n n p + 1 q X L j n,l + R(n,p )=0 L2N() X L j p,l + R(n,p )=0 L2N() x x L L

29 Finite Volume Discretisation X L (x L x ) ("r )+q (C + p n n p + 1 q X L j n,l + R(n,p )=0 L2N() X L j p,l + R(n,p )=0 L2N() x x L L

30 How to choose the fluxes? Desired properties: stable preservation of max principle approximate boundary layers well consistency with thermodynamic equilibrium

31 Consistency with equilibrium Zero current leads to 0 = r U T r n

32 Consistency with equilibrium Zero current leads to 0 = r U T r n Mimic this numerically via with 0= U T n = L and = L

33 Consistency with equilibrium Zero current leads to 0 = r U T r n Mimic this numerically via Important for coupling! with 0= U T n = L and = L

34 Finite Volume Discretisation X L (x L x ) ("r )+q (C + p n n p + 1 q X L j n,l + R(n,p )=0 L2N() X L j p,l + R(n,p )=0 L2N() constant!

35 How to choose the fluxes? Scharfetter & Gummel (1969) for Boltzmann statistics: d dx j n,l =0 with n(x )=n n(x L )=n L

36 How to choose the fluxes? Scharfetter & Gummel (1969) for Boltzmann statistics: d dx j n = d dx qµ U T d dx n n d dx =0 with n(x )=n n(x L )=n L

37 How to choose the fluxes? Scharfetter & Gummel (1969) for Boltzmann statistics: d dx j n = d dx qµ U T d dx n n d dx =0 with n(x )=n n(x L )=n L leads to exact solution j = qµu T "B L U T n L B L U T n #,

38 How to choose the fluxes? Scharfetter & Gummel (1969) for Boltzmann statistics: d dx j n = d dx qµ U T d dx n n d dx =0 with n(x )=n n(x L )=n L leads to exact solution j = qµu T "B L U T n L B L U T n #, B(x) := x e x 1 B(x) Bernoulli function

39 How to choose the fluxes? Scharfetter & Gummel (1969) for Boltzmann statistics: d dx j n = d dx qµ U T d dx n n d dx =0 with n(x )=n n(x L )=n L leads to exact solution j = qµu T "B L U T n L B L U T n #, B(x) := x e x 1 Exact solution for Boltzmann statistics! What about other? F 0.5 B(x) Bernoulli function

40 How to choose the fluxes? Scharfetter & Gummel (1969) for Boltzmann statistics: d dx j n = d dx qµ U T d dx n n d dx =0 with n(x )=n n(x L )=n L leads to exact solution j = qµu T "B L U T n L B L U T n #, B(x) B(x) := x e x 1 Bernoulli function Exact solution for Boltzmann statistics! What about other? F Three flux approx. for Blakemore!

41 General Scharfetter-Gummel oprucki/gärtner (2013): d dx j n = d dx qµ U T g n N c d dx n n d dx =0 with n(x )=n n(x L )=n L

42 General Scharfetter-Gummel oprucki/gärtner (2013): d dx j n = d dx qµ U T g n N c d dx n n d dx =0 with n(x )=n n(x L )=n L Exact solution for Blakemore j = j 0 ng nl N C B L U T + j nl j 0 N C g n N C B L U T o j n j 0 NC with g(x) = x(f 1 ) 0 (x) = 1 1 x and j 0 = qµn C U T

43 General Scharfetter-Gummel oprucki/gärtner (2013): d dx j n = d dx qµ U T g n N c d dx n n d dx =0 with n(x )=n n(x L )=n L Exact solution for Blakemore j = j 0 ng nl N C B L U T + j nl j 0 N C g n N C B fixed point equation exact (up to machine precision) L U T o j n j 0 NC with g(x) = x(f 1 ) 0 (x) = 1 1 x and j 0 = qµn C U T

44 Diffusion averaging Bessemoulin-Chatard (2012) and oprucki et al. (2014): d dx j n = d dx qµ U T g n N c d dx n n d dx =0 with n(x )=n n(x L )=n L

45 Diffusion averaging Bessemoulin-Chatard (2012) and oprucki et al. (2014): "B j = qµu T ḡ L L n U T ḡ L B L U T ḡ L n L #

46 Diffusion averaging Bessemoulin-Chatard (2012) and oprucki et al. (2014): "B j = qµu T ḡ L L n U T ḡ L B L U T ḡ L n L # where ḡ L = F 1 nl N c F 1 n Nc log(n L /N c ) log(n /N c ) only consistent average! approximates g(x) =x(f 1 ) 0 (x) = (F 1 ) 0 (x) log 0 (x)

47 Inverse Activity Coefficients Fuhrmann (2015): d dx j n = d qµn c dx ( )e d dx ' n =0 with (x )= (x L )= L with ( ) = F( ) e leads to

48 Inverse Activity Coefficients Fuhrmann (2015): d dx j n = d qµn c dx ( )e d dx ' n =0 with (x )= (x L )= L with ( ) = F( ) e leads to j = qµu T N c L e L B L U T e k B L U T

49 Inverse Activity Coefficients Fuhrmann (2015): d dx j n = d qµn c dx ( )e d dx ' n =0 with (x )= (x L )= L with ( ) = F( ) e leads to j = qµu T N c L e L B L U T e k B consistent for any average! L U T where L = ( )+ ( L ) 2 2 [ ( ), ( L )]

50 Compare three schemes General Scharfetter-Gummel j GENSG = j 0 e L B L + j j 0 e B L j j 0 Diffusion enhanced Scharfetter-Gummel j DESG = j 0 g L F( L )B L g L F( )B L g L Inverse activity based scheme j IACT = j 0 L ne L B ( L ) e B ( L ) o

51 How to design a benchmark? Leave equilibrium! ' L = L + L r' = r + r

52 How to design a benchmark? Leave equilibrium! ' L = L + L r' = r + r h L L (linear potential) coarse grid large bias

53 How to design a benchmark? Leave equilibrium! ' L = L + L r' = r + r large contrast with opposite sign h L L (linear potential) coarse grid large bias = F 1 (n) large density contrast

54 Errors between schemes diffusion enhanced vs general SG inverse act. vs general SG

55 Errors between schemes L = L diffusion enhanced vs general SG inverse act. vs general SG

56 Errors between schemes L = L same contour line diffusion enhanced vs general SG inverse act. vs general SG

57 Errors between schemes diffusion enhanced vs general SG inverse act. vs general SG

58 pin N I =0.00/cm 3 N A = /cm 3 N D = /cm 3 0.5µm 2µm 2µm 2µm How do schemes influence current and electrostatic potential?

59 IV curves current [A] refinement level: 1 reference solution diffusion enhanced SG generalized SG inverse activities max error current [A] 1x10-2 1x10-3 1x10-4 1x10-5 1x10-6 1x10-7 max errors IV curves h 2 diffusion enhanced SG generalized SG inverse activities bias [V] h

60 Results three thermodynamically consistent schemes (Blakemore) all schemes converge to SG for large negative pin benchmark useful to discriminate accuracy exact scheme yields best current approximation; diffusion enhanced scheme good for electrostatic field other factors: computation times need to be considered

61 Outlook prototype: ddfermi moderately-sized 2D/3D problems different variables: heterostructure we welcome applications!

62 Muchas gracias por su atención! Thank you for your attention!