Report

Mobility and Diffusion under the Premise of Solar Cells The Role of Energy-Transport Dan Mendels, Nir Tessler Sara & Moshe Zisapel Nanoelectronic Center Electrical Engineering Dept. Haifa 32000 Israel www.ee.technion.ac.il/nir The operation of Solar Cells is all about balancing Energy Think “high density” or “many charges” NOT “single charge” There is extra energy embedded in the ensemble If you came from session P. There is also pseudo band like behavior The Physical Framework • Steady State I-V measurements • Steady State Qausi Equilibrium (Incl. Traps) [1] K. C. Kao and W. Hwang, Electrical transport in solids vol. 14. New York: Pergamon press, 1981. [2] H. T. Nicolai, M. M. Mandoc, and P. W. M. Blom, "Electron traps in semiconducting polymers…" PRB, 83, 195204, 2011. • Not the transient, possibly dispersive, transport where D/m may be VERY HIGH R. Richert, L. Pautmeier, and H. Bassler, "Diffusion and drift of charge-carriers in a random potential - deviation from Einstein law," Phys. Rev. Lett., vol. 63, pp. 547-550, 1989. Original Motivation Measure Diodes I-V Extract the ideality factor Y. Vaynzof et. al. JAP, vol. 106, p. 6, Oct 2009. The ideality factor Is the Generalized Einstein Relation The Generalized Einstein Relation is NOT valid for organic semiconductors G. A. H. Wetzelaer, et. al., "Validity of the Einstein Relation in Disordered Organic Semiconductors," PRL, 107, p. 066605, 2011. Monte-Carlo simulation of transport d 0 dx Standard M.C. means uniform density Charge Density relative to DOS 0.05 10-4 10-3 10-2 Y. Roichman and N. Tessler, "Generalized Einstein relation for disordered semiconductors - Implications for device performance," APL, 80, 1948, 2002. Einstein Relation [eV] G.E.R. 0.04 0.03 0.02 Monte-Carlo 0.01 0 17 10 18 10 19 10 Charge Density [1/cm3] 20 10 Comparing Monte-Carlo to Drift-Diffusion & Generalized Einstein Relation Implement contacts as in real Devices d 0 dx 19 4 10 3 Carrier Density [1/cm ] 3 Carrier Density [1/cm ] 19 2.5 10 19 2 10 1.5 1019 qE 1 1019 5 1018 0 19 3.5 10 19 3 10 2.5 1019 2 1019 qE 1.5 1019 1 1019 GER Holds for real device5 Monte-Carlo Simulation 1018 0 20 40 60 80 100 Distance from 1st lattice plane [nm] 0 0 20 40 60 80 100 Distance from 1st lattice plane [nm] Where does most of the confusion come from D The intuitive Random Walk d The coefficient describing dx d J e qnme E qDe n dx Generalized Einstein Relation is defined ONLY for J. Bisquert, Physical Chemistry Chemical Physics, vol. 10, pp. 3175-3194, 2008. E d What is Hiding behind dx E There is an Energy Transport X X Charges move from high density region to low density region Charges with High Energy move from high density region to low density The Energy Balance Equation dn dE J qnm F nqD m ddx x The operation of Solar Cells is all about balancing E nergy R DE How much “Excess” energy is there? Energy [] 150meV Distribution [a.u.] Distribution [a.u.] 1.2 -5 1 0.8 0.6 -4 -3 -2 -1 0 1 Density Of States =3kT; T=300K 0.4 0.2 0 -0.4 1 -0.3 -0.2 -0.1 0 Jumps DN Energy [eV] Jump UP Carriers 0.8 =3kT DOS = 1021cm-3 N=5x1017cm-3=5x10-4 DOS Low Electric Field E 0.6 0.4 0.2 0 -0.4 -0.3 -0.2 -0.1 0.1 0 0.1 Energy [eV] B. Hartenstein and H. Bassler, Journal of Non - Crystalline Solids 190, 112 (1995). The High Density Picture Mobile and Immobile Carriers Distribution [a.u.] 1 Carriers Jumps distribution 0.8 Is it a BAND? 0.6 0.4 0.2 0 -0.4 Mobile Carriers -0.3 -0.2 -0.1 =3kT DOS = 1021cm-3 N=5x1017cm-3=5x10-4 DOS Low Electric Field 0 0.1 Energy [eV] Transport is carried by high energy carriers Summary • Transport: Many Charges Thank You ≠ Single Charge – Mobile and Immobile (“trapped”, “Band”) charges • Transport of energy! – There is “excess” energy in the system. dE dn J qnm F nm qD dx dx dE E n E T dx n x T x • Where do the carriers hop in energy – Not around EF. • Ideality factor Einstein relation? Seebeck Effect Mott’s Variable Range Hopping E R A D* DE A* D DE Rexp k BT R 0e R R DE * n C8H17 C8 H17 N N * DE * n* C8 H1 7 S PFOBT 2r e DE K BT C8 H17 InGaAs InP PFO r r and DE are determined so as to maximize the hopping rate For a shaped density of states: For a constant density of states: e Transport Energy (Et=?) E ? DE 4 3 r DE 1 3 Effective intermediate energy Effective initial energy r a shaped density of states: 1. Mobility is charge density dependent e Transport Energy (Et=?) DE E ? Re Effective intermediate energy Et E K BT Effective initial energy 2. E EF 0.000 -0.005 -0.010 -0.015 -0.020 -0.025 16 10 3. E is E (n, T ) KBT t E [eV] Transport Energy 17 18 10 10 There is transport of energy even in the absence of Temperature gradients 19 10 -3 Charge Density [cm ] Effective Initial Energy E J qnm ( x) F ( x) qD( x) (a) E EF -0.1 -0.15 Effective -0.2 -0.25 Average -0.3 q-Fermi 10 16 10 17 10 18 10 19 10 3 Charge Density [1/cm ] 20 Energy [eV] -0.05 dn dx 1 dE dn J qn( x) m ( x) F ( x) q D( x) q dx dx What if we analyze the standard (uniform density) Monte-Carlo Charge Density relative to DOS -4 Einstein Relation [eV] 0.05 10 -3 10 -2 10 GER 0.04 d 0 dx 0.03 Monte-Carlo 0.02 0.01 0 1017 1018 1019 Charge Density [1/cm3] 1020 e- & E Does the Generalized Einstein Apply Does your system obey the laws of Thermodynamics