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Simulated annealing electro-photonic optimization of organic solar cells

Christoph Kirscha) and Sorin Mitranb)

Department of Mathematics, University of North Carolina, Chapel Hill, North Carolina 27599-3250, USA

(Received 21 March 2012; accepted 25 July 2012; published online 6 September 2012)

Micro-patterned organic solar cells can exhibit enhanced light absorption properties due to a photoniccrystal effect [Tumbleston et al., Appl. Phys. Lett. 94, 043305 (2009)]. Here, a three-dimensionalmodel of light absorption and charge carrier transport in micro-patterned materials is presented andapplied to the design of organic bulk heterojunction (BHJ) solar cells. Rigorous coupled waveanalysis is used to simulate the multiple scattering and absorption of light in a layered solar celldevice. The non-linearly coupled steady-state electric field and charge transport equations are solvediteratively by a sequence of linear partial differential equations (PDEs). Each linear PDE isdiscretized by an exponential upwind finite difference scheme, and the preconditioned conjugategradient method is applied to the resulting algebraic system. The electro-photonic solver is coupledwith the simulated annealing optimization algorithm to investigate the effect of micro-patterningupon performance of BHJ solar cells. Starting from the baseline configuration of a cell formed fromflat layers of optimal thickness, the optimization algorithm leads to improvements of up to 15%in energy conversion efficiency by patterning both the front and back electrodes with a periodicridge shape, with conformally coated layers in-between. VC 2012 American Institute of Physics.[http://dx.doi.org/10.1063/1.4748314]

I. INTRODUCTION

Photovoltaic cells are typically made of layers of differ-ent materials. One-dimensional mathematical models1 maycapture phenomena in cells consisting of flat layers, but thedevice simulation of solar cells with a spatial variation ofmaterial properties in the directions orthogonal to the direc-tion of the incident light requires higher-dimensional models.Such variation of material parameters is interesting, becausea periodic pattern with a characteristic length of 10!7 m (i.e.,on the order of the wavelength of light) may lead toenhanced light absorption in the photoactive layer, comparedto a flat configuration.2 As a result, the electron-hole pairgeneration rate in the solar cell will be higher and a largercurrent—and thus higher efficiency—is expected. Such geo-metric performance enhancement complements materialdesign improvements, and needs to be investigated fromboth the optics and electronics point-of-views.

We shall present three-dimensional mathematical mod-els for the light scattering and absorption, as well as for thecharge carrier transport in a patterned organic bulk hetero-junction (BHJ) solar cell. These models are coupled via theexciton generation rate density, which is computed from theoptics simulation and used as a source term in the chargetransport simulation. For a given shape of the material layers,our multiphysics simulation will yield the current-voltagecharacteristics of the corresponding solar cell. In particular,key figures such as the short circuit current, open circuit volt-age, and maximum power point can be determined, fromwhich the fill factor of the solar cell and also its energy con-version efficiency may be computed.

The simulated annealing algorithm3,4 is used to maximizethe energy conversion efficiency by varying the shape of thesolar cell device. This allows us to identify optimal layer thick-nesses for flat cells with given material properties, as well asoptimal patterns. We also analyze the potential for furtherimprovement and identify the main loss mechanisms remain-ing after shape optimization. The research code (written in C)developed to do the simulations is publicly available.5

This paper is organized as follows: In Sec. II, we reviewthe solution of the time-harmonic Maxwell’s equations tocompute the electric field by rigorous coupled wave analy-sis,6,7 in a multi-layered solar cell device. Post-processing ofthe numerical solution yields the exciton generation rate den-sity everywhere in the photoactive layer. In Sec. III, wedescribe the charge transport in the organic bulk heterojunc-tion material based on the classical semiconductor equa-tions,8 adopting an effective medium approach. We alsodiscuss the transformation to dimensionless variables9 andderive an expression for the net charge carrier generationrate, which includes electric field-dependent exciton dissoci-ation.10,11 From the numerical solution, current-voltage char-acteristics of the solar cell device may be determined, and inparticular its energy conversion efficiency can be computed.In Sec. IV, we combine our multiphysics simulation with thesimulated annealing metaheuristic to identify optimal shapes.While we find no discernible performance enhancementfrom micro-patterning of the bulk-heterojunction layer only,combined patterning of multiple layers can lead to a 15%improvement in power per unit area. Numerical implementa-tion details are presented in the appendices.

II. LIGHT SCATTERING, ABSORPTION, AND EXCITONGENERATION

When light enters a solar cell device, it is scattered atthe various interfaces and partially absorbed in the material

a)ckirsch@email.unc.edu.b)mitran@unc.edu.

0021-8979/2012/112(5)/054502/13/$30.00 VC 2012 American Institute of Physics112, 054502-1

JOURNAL OF APPLIED PHYSICS 112, 054502 (2012)

layers (Fig. 1). The exciton generation rate densityG "m!3s!1# is computed as an integral involving the time-averaged power density of absorbed light in the photoactivelayer, hQix "Wm!3#, over the angular frequency x :$ 2pc=k,where k "m# denotes the wavelength, and where c denotes thespeed of light in vacuum

G :$!1

0

hQix!hx

dx; hQix :$ x2!0 Im%!r;x&jExj2: (1)

The quantities G, hQix, the complex relative permittivity!r;x, and the time-harmonic electric field Ex "Vm!1# vary inspace; !h denotes the reduced Planck constant and !0 denotesthe vacuum permittivity. Equation (1) means that eachabsorbed photon generates one exciton, i.e., we have notincluded any multiple exciton generation from high-energyphotons, an effect which has been observed in quantum dots.12

The electric field Ex in the device satisfies the second-order time-harmonic Maxwell equations

curl curl Ex $x2

c2!r;x Ex; x > 0: (2)

The complex relative permittivity !r;x $ %nx ' ijx&2, withi2 $ !1, has been measured13 in the range of wavelengthsfrom 400 nm to 700 nm for all materials present in the solar celldevice (Fig. 2). Equations of the form (2) must be solved formany different wavelengths in order to approximate the integralin Eq. (1). The structure of the solar cell device—layered in thedirection of the incident light and periodic in the orthogonaldirections—motivates the use of rigorous coupled wave analy-sis6,7 to solve (2), instead of a full-wave simulation. We reviewthis method in the remainder of this section; details on thenumerical implementation are given in Appendix A.

A. Series expansion in terms of plane waves, Fouriertransform, generalized eigenvalue problem

Throughout this section, we consider a single fixedangular frequency x > 0, and we omit all indices x to sim-plify notation. We restrict our considerations to light propa-gating in the z-direction (Fig. 1), and we assume discrete

translational invariance of Eq. (2) on a lattice K ( R2 in thexy-plane orthogonal to the direction of propagation. Thisalso requires all interfaces present in the solar cell device tobe either parallel or perpendicular to the xy-plane; if multipledirections of propagation are to be considered, an integralover the first Brillouin zone is also involved.14 The coeffi-cients in a series expansion of E, in terms of plane waves in(x, y)-directions, are given by

EG%z& $1

jX0j

!

X0

e!iG)rE%r; z& dr; (3)

with r :$ %x; y&> 2 X0, where X0 denotes the primitive cellof the lattice K, containing the origin, and where G 2 K0 isany vector in the reciprocal lattice: eiG)R $ 1; 8R 2 K. Thisreduction to X0 *R is due to Bloch’s theorem.15 With Max-well’s equations (2), we obtain a system of linear ordinarydifferential equations (ODEs) for EG $ %Ex

G;EyG;E

zG&>

$: %E?G ;EkG&>. The coefficients in these ODEs,

!r;G;G0%z& $1

jX0j

!

X0

e!i%G!G0&)r!r%r; z& dr; (4)

FIG. 1. A conventional organic solar cell device with four flat layers of finitethickness (ITO, Poly(3,4-ethylenedioxythiophene) poly(styrenesulfonate)(PEDOT:PSS), P3HT:PCBM bulk heterojunction, nc-ZnO) and two semi-infinite layers (glass, Al). Incoming light is scattered at the material interfacesand partially absorbed in the photoactive layer (3). Excitons (7) are generatedthere which dissociate into free electrons (–) and holes (') that are transportedto the electrodes (anode: 1, 2, cathode: 4, 5) by an electric field, !rw.

FIG. 2. Measured optical constants13 for the six materials used in the solarcell device. Refractive index n (a) and extinction coefficient j (b) vs. wave-length k "nm#. The extinction coefficients of glass and nc-ZnO are zero in therange of wavelengths considered here, and therefore not shown.

054502-2 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

G;G0 2 K0 are piecewise constant due to the assumed lay-ered structure of the solar cell device in the z-direction. Thusin each layer, the linear ODEs may be solved using the Fou-rier transform. For the series expansion coefficients of the or-thogonal field components, E? :$ fE?GgG2K0 , we then obtain

E?%z& $X1

j$1

%A'j eikjz ' A!j e!ikjz&E?j ; (5)

where the pairs %kj; E?j &; j 2N are solutions of a generalizedeigenvalue problem of the form

ME? $ k2NE?: (6)

The complex matrices M $ fMG;G0 gG;G02K0 and N $fNG;G0 gG;G02K0 consist of 2* 2 blocks with entries dependingon the angular frequency x and on the layer values of !r;G;G0

(4). The square root of the generalized eigenvalues k2j is

taken such that arg kj 2 %!p=2; p=2#; with this choice of kj,the “forward” partial waves A'j e!Im%kj&zei Re%kj&zE?j in Eq. (5)travel in the positive z-direction, whereas the “backward”partial waves A!j eIm%kj&ze!i Re%kj&zE?j travel in the negativez-direction—they are evanescent if Im%kj& > 0.

Once the solution pairs %k2j ; E?j & of Eq. (6) have been

identified in each layer, it remains to match the solutions (5)across the interfaces and with the boundary conditions, inorder to find the values of the amplitudes A6

j in each layer.For that purpose, the orthogonal components of the seriesexpansion coefficients of the magnetic field B $ l0H "T#(assuming non-magnetized materials with vacuum perme-ability l0) are also used. The coefficients H? :$ fH?GgG2K0

are related to E? via the time-harmonic Maxwell-Faradayequation: curl E $ i xl0H.

Consider M > 0 layers of finite thickness with two addi-tional, semi-infinite layers on the front and back, so that theinterfaces are located at the vertical positions z0;…; zM.These interfaces coincide with the material interfaces in theflat case (Fig. 1), but not in general (Fig. 3). The orthogonal

components of both the electric and magnetic fields must becontinuous across the interfaces between layers. DefiningY :$ %E?;H?&>, we have in the mth layer

Ym%z& $ ~F mei ~K mz ~Am; m $ 0;…;M ' 1; (7)

with a complex matrix ~Fm containing the eigenvectors E?m;j,a diagonal complex matrix ~Km containing the eigenvalues6km;j, and with a complex vector ~Am containing the scatter-ing amplitudes A6

m;j. The tilde above these quantities indi-cates permutation, i.e., a change in the order of summationin Eq. (5). The choice of this permutation is crucial for thenumerical stability of the transfer matrix method, as dis-cussed in Sec. II B.

B. Transfer matrix method

The matching conditions at the interfaces,

Ym%zm& $ Ym'1%zm&; m $ 0;…;M; (8)

yield a system of M' 1 linear equations for the M' 2 scat-tering amplitudes A6

m ; m $ 0;…;M ' 1. With Eq. (7) wealso have

ei ~K mdmYm%zm!1& $ Ym%zm&; m $ 1;…;M; (9)

where dm :$ zm ! zm!1 > 0 denotes the thickness of the mthlayer.

The forward scattering matrices S0;m relate A60 and A6

m .They can be computed recursively starting from S0;0 $ I(identity matrix), by alternate application of transfer andpropagation steps (8) and (9). A similar recursion may beused to compute the backward scattering matrices Sm;M'1

relating A6m and A6

M'1. The permutation mentioned inSec. II A is used to prevent numerical instability duringrepeated application of propagation steps (9), when M islarge.16 The appropriate permutation exchanges the positionsof km;j and !km;j in ~Km if Im%km;j& < 0.

With the scattering matrices, the system of linear equa-tions (8) can be transformed into block-diagonal form,17 sothat the unknown amplitudes in each finite layer, A6

m ;m $ 1;…;M, as well as the reflection, A!0 , and the transmis-sion, A'M'1, are finally related to the boundary valuesA'0 ; A!M'1 (Sec. II C) via M' 1 equations.

C. Incoming sunlight and radiation condition

The incoming light is modeled as a plane wave comingfrom !1 and propagating in the positive z-direction. One-sun illumination is simulated by use of the AM1.5 referencesolar spectral irradiance.18 We assume that the electric per-mittivity in the front semi-infinite layer is a positive con-stant: !r%r; z& $ !r;0 > 0; z < z0 (Fig. 1). Then, the matricesM; N in the generalized eigenvalue problem (6) are blockdiagonal, and we obtain Einc

G + 0; G 6$ 0, and

Einc0 %z& $

A'0;j1

A'0;j20

0

B@

1

CAeikz; k $ xc

"""""""!r;0p

> 0 (10)

FIG. 3. A solar cell device with patterned front and back electrodes and con-formally coated layers. Also indicated are the interfaces used in the opticssimulation (Sec. II), the computational domain X and boundaries R6; Rp

used in the charge transport simulation (Sec. III C), as well as the geometryparameters used in the shape optimization (Sec. IV).

054502-3 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

with A'0;j1;2 2 C. Here, j1; j2 are the indices of eigenbasis vec-tors (solutions of Eq. (6)) with E?j1;0 $ %1; 0&

> andE?j2;0 $ %0; 1&

>. The two non-zero components A'0;j1;2 2 C ofthe vector A'0 characterize the polarization of the incominglight, and they are only constrained by the prescribed irradi-ance18 I "Wm!2# for the angular frequency x

jA'0;j1 j2 ' jA'0;j2 j

2 $ 2xl0

kI: (11)

In order to incorporate some of the unpolarized nature ofactual sunlight into the simulation, we take an average overfield computations for several linear polarizations

A'0;j1 $"""""""""""""2xl0

kI

rcos h; A'0;j2 $

"""""""""""""2xl0

kI

rsin h; (12)

where h denotes the angle of the polarization plane withrespect to the x-axis.

For the other boundary value, we use A!M'1 $ 0. Thiscorresponds to imposing a radiation condition at infinity, i.e.,there should be no waves coming from '1 and propagatingin the negative z-direction.

D. Exciton generation rate density, light absorptionefficiency

When the scattering amplitudes have been computed,the exciton generation rate density G is evaluated at everypoint (r, z) in the photoactive material with (1) and (3)

G%r; z& $ !0

2!h

X

G;G02K0ei%G!G0&)rIBHJ

G;G0%z&; (13)

where the z-dependence involves an integral over the angularfrequency x

IBHJG;G0%z& :$

!1

0

Im%!BHJr;x &Ex;G%z& ) Ex;G0%z& dx; (14)

for G;G0 2 K0. In Eq. (14), !BHJr;x denotes the complex relative

permittivity of the photoactive (bulk heterojunction) mate-rial. Because the imaginary part, Im%!BHJ

r;x & $ 2nBHJx jBHJ

x , hascompact support (Fig. 2), the integral in Eq. (14) involvesonly a finite interval in a practical computation. The parallelelectric field components Ekx;G in Ex;G $ %E?x;G;E

kx;G&

> arecomputed from the orthogonal magnetic field componentsH?x;G using Ampere’s circuital law, curl Hx $ !i x!0!r;xEx.Thus, no further derivatives are required, and all field com-ponents necessary to evaluate the exciton generation rate(13) and (14) in the photoactive part of the mth layerfzm!1 < z < zmg (Fig. 3) are contained in Ym (7),m $ 1;…;M. Other quantities of interest, which are notrequired for the coupling with the charge transport simula-tion (Sec. III), can also be computed from the optics simula-tion results, such as the reflection, transmission, andabsorption in each material (Fig. 4).

We denote the domain occupied by the photoactive ma-terial by X ( X0 *R. The power per unit area of lightabsorbed in the photoactive material is given by

Pabs :$ 1

jX0j

!

X

!1

0

hQix%x& dx dx "Wm!2#: (15)

With the reference solar spectral irradiance18 Ik (for wave-lengths k , 400 nm), we also compute the light absorptionefficiency of the photoactive region

gabs :$ Pabs

Pin; Pin :$

!1

400 nm

Ik dk ’ 954 Wm!2: (16)

III. CHARGE TRANSPORT, EXCITON DISSOCIATION,AND CHARGE CARRIER RECOMBINATION

We use the exciton generation rate density G computedin Sec. II as a source term in the charge transport simulation.The photoactive material in an organic solar cell device isregarded as an effective insulator, with a band gap Egap "J#given by the difference between the lowest unoccupied mo-lecular orbital (LUMO) level of the electron acceptor mate-rial (such as [6,6]-phenyl-C61-butyric acid methyl ester(PCBM)) and the highest occupied molecular orbital(HOMO) level of the electron donor material (such asP3HT).19 Electric current is produced as photogeneratedexcitons dissociate into free charge carriers and are trans-ported to the electrodes by an electric field. For the chargetransport in patterned solar cell devices such as the one illus-trated in Fig. 3, we extend a one-dimensional metal-insula-tor-metal model1 to three space dimensions.

FIG. 4. Relative reflection and absorption in each material, obtained fromthe optics simulation (Sec. II). (a) Flat benchmark cell (gabs $ 0:288) and(b) double ridge pattern with maximum energy conversion efficiency(gabs $ 0:329) (Sec. IV). Percentage values in legends state the cumulativerelative reflection/absorption in the window 400 nm - k - 659 nm, whereP3HT:PCBM is absorbing (Fig. 2).

054502-4 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

We note that the model described below includes chargetransport in the photoactive material only. An approach toincorporate the non-active materials into the charge transportmodel as effective resistors has been proposed recently.20

A. Coupled charge carrier transport equations

The semiconductor equations8 involve the electric poten-tial w "V# and the charge carrier number densities n, p "m!3#of electrons and holes, respectively. These coupled partial dif-ferential equations (PDEs) describe the drift and diffusion ofeach charge carrier species in the material at the continuumscale, and they also relate the total charge density to the elec-tric field. The steady-state semiconductor equations in non-doped material are given by

div %!rw& $ q%n! p&; (17)

div Jn $ U%jrwj; n; p&; Jn :$ nlnrw! Dnrn; (18)

div Jp $ U%jrwj; n; p&; Jp :$ !plprw! Dprp: (19)

In Eqs. (17)–(19), the (static) electric permittivity and the ele-mentary charge are denoted by ! "Fm!1# and q "C#, respectively.The charge carrier mobilities ln;p "m2%Vs&!1# and diffusioncoefficients Dn;p "m2s!1# satisfy the Einstein relation20

Dn;p $ ln;pVt; Vt :$ kBT

q"V#; (20)

where the thermal voltage Vt involves the Boltzmann con-stant kB "JK!1# and the absolute temperature, T [K]. The netcharge carrier generation rate density U "m!3s!1# is dis-cussed in Sec. III B, and the boundary conditions are pre-sented in Sec. III C.

At thermal equilibrium, the electron and hole fluxesJn; Jp "m!2s!1# vanish, and we find exponential relationshipsbetween the charge carrier number densities n, p and theelectric potential w. This motivates a transformation todimensionless variables (u, v, w), which are related to thenatural variables %w; n; p& via9

w $ uVt; n $ Nintveu; p $ Nintwe!u: (21)

In Eq. (21), the intrinsic charge carrier number density Nint

"m!3# depends on the effective densities of statesNc; Nv "m!3# in the conduction and valence band, respec-tively, as well as on the band gap; it is defined by

Nint :$""""""""""NcNv

pexp ! 1

2

Egap

qVt

# $: (22)

After division by Nint, the steady-state semiconductor equa-tions (17)–(19) take the form F%u& $ 0, where u :$%u; v;w&> and F :$ %Fu;Fv;Fw&> with

Fu%u& :$ div %k2Dru& ! veu ' we!u; (23)

Fv%u& :$ div %Dneurv& ' ~U%jruj; v;w&; (24)

Fw%u& :$ div %Dpe!urw& ' ~U%jruj; v;w&; (25)

where ~U :$ U=Nint "s!1#, and where the characteristic Debyelength kD "m# is given by

kD :$

""""""""""!Vt

qNint

s

: (26)

A number of monographs cover semiconductor devicephysics and modeling;21,22 for a mathematical analysis ofEqs. (17)–(19), we refer the reader to Jerome.23,24 For thetime-dependent version of Eqs. (17)–(19), exponential con-vergence to the unique steady state has been shownanalytically.25,26

B. Net charge carrier generation rate

Photogenerated excitons may dissociate into free chargecarriers (electron-hole pair) if a donor/acceptor interface islocated within a distance of the order of the exciton diffusionlength (.10!8 m) from the point of generation. In a polymer-fullerene blend such as P3HT:PCBM, the characteristic lengthscales of both phases are comparable to the exciton diffusionlength,27 and therefore these blends are interesting candidatesfor the photoactive layer in a solar cell device.

Thus, we neglect the exciton diffusion in the photoactivelayer, and we consider the following local generation andloss mechanisms

groundstate

G!! kf X

excitonsX

kdissX!! R

free chargecarriers n; p:

(27)

In Eq. (27), X "m!3# denotes the exciton number density, Gdenotes the exciton generation rate density (Sec. II), R"m!3s!1# denotes the charge carrier recombination rate den-sity, and kdiss; kf "s!1# denote the exciton dissociation anddecay rates, respectively. At steady state, these rate densitiesmust be balanced

%kdiss ' kf &X $ G' R: (28)

Therefore, we may obtain the net charge carrier generationrate density U from a convex combination of exciton genera-tion and charge carrier recombination

U $%27&kdissX ! R $%28&

PdissG! %1! Pdiss&R (29)

with the dissociation probability (the fraction of excitonswhich dissociates into free charge carriers)

Pdiss :$ kdiss

kdiss ' kf: (30)

The bimolecular model28 is commonly used for the recombi-nation rate density R

R :$ c%np! N2int&; c $ q

!min ln; lp

% &: (31)

Several different expressions have been proposed for theLangevin recombination constant c "m3s!1# in the case of

054502-5 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

polymer-fullerene blends.29,30 The exciton dissociation ratekdiss "s!1# depends on the electric field strength and on theelectron-hole pair separation distance x > 0 "m#, and is givenby10,11

kdiss%jrwj; x& $ 3c4px3

exp !Eb%x&qVt

# $I1%2b&

b: (32)

Equation (32) involves the first-order modified Bessel func-tion of the first kind, I1

I1%2b&b$X1

k$0

b2k

k!%k ' 1&!; b :$

"""""""""""""qjrwj4p!V2

t

s

; (33)

and the coulombic electron-hole pair binding energy is givenby Eb%x& :$ q2=%4p!x& "J#.

The disorder in the polymer-fullerene blend is taken intoaccount by assuming a random electron-hole pair separationdistance with a Maxwell-Boltzmann distribution31

f %x; a& :$ 4x2

"""pp

a3exp ! x2

a2

# $; x > 0: (34)

The parameter a > 0 "m# is the location of the maximumof the probability density function f % ) ; a&; the expected valueof the electron-hole pair separation distance is given by2a=

"""pp

. Then, the exciton dissociation rate kdiss and thus thedissociation probability (30) also become random variables.The value used for Pdiss in Eq. (29) is the expected value

Pdiss%jrwj; a; kf & :$!1

0

kdiss%jrwj; x&kdiss%jrwj; x& ' kf

f %x; a& dx: (35)

All material properties of P3HT:PCBM which we have usedin our simulations are listed in Table I.

C. Boundary conditions

The semiconductor equations (17)–(19) have to besolved in the domain X occupied by the polymer-fullereneblend. This domain is bounded by the electrodes and by thefaces of the right prism X0 *R, where X0 denotes the primi-tive cell of the lattice used in the optics simulation(Sec. II A). The boundary of X is decomposed into@X $ R' [ R! [ Rp, where R' and R! denote the cathodeand anode boundaries, respectively (Fig. 3). We impose peri-

odic boundary conditions at Rp; the boundary conditionsimposed at R6 are specified below.

We still use the metal-insulator-metal picture, so thatthe following five positive parameters describe the energylevels in the materials (Fig. 5(a)): the electron affinity v, thework function /, and the band gap Egap of the insulator, aswell as the work functions /a and /c of the anode and cath-ode, respectively, with /c < /a. When the thermal equilib-rium is established after contact, the Fermi levels arealigned, and Ec; Ev vary across the insulator accordingly(Fig. 5(b)). Upon applying an external voltage Va "V# (bias),the thermal equilibrium is disturbed and we consider quasi-Fermi levels EFn ; EFp for the electrons and holes, respec-tively (Fig. 5(c)). They are assumed to be pinned to the metal

TABLE I. Material parameters used for the charge transport simulation in

P3HT:PCBM, with references to the equations where they first appear.

! 3:4 !0 Fm!1 (17)

ln 2* 10!7 m2%Vs&!1 (18)

lp 10!8 m2%Vs&!1 (19)

T 300 K (20)""""""""""NcNvp

2:5* 1025 m!3 (22)

Egap 0.95 eV (22)

kf 2* 104 s!1 (27)

a 1:8* 10!9 m (34)

FIG. 5. Illustration of the energy levels in the metal-insulator-metal modelused for the charge transport simulation: the five parameters v; Egap; /;/a; /c describing the energetic properties of the metals and of the insulator(a), the thermal equilibrium after contact at R6 (b), and the situation underforward bias (c). The band bending near the contacts is not shown in theseillustrations.

054502-6 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

Fermi levels at each boundary and to be separated across theinsulator by the applied voltage

EFn jR! $ EFp jR! ; EFn jR' $ EFp jR' ; (36)

%Ei ! EFp&jR! $ /a ! /; (37)

%EFn ! Ei&jR' $ /! /c; (38)

EFn jR' ! EFp jR! $ qVa: (39)

Equations (36)–(39) translate to Dirichlet boundary condi-tions for the unknowns %w; n; p& if we assume a Boltzmanndistribution for the occupation probability of electron statesat the boundaries32

wjR' ! wjR! $1

q%/a ! /c& ! Va; (40)

p

Nint

'''R!$ Nint

n

'''R!$ exp

/a ! /qVt

# $; (41)

n

Nint

'''R'$ Nint

p

'''R'$ exp

/! /c

qVt

# $: (42)

The unspecified additive constant in the electric poten-tial w may be neglected because the semiconductor equations(17)–(19) involve only the gradient, rw; therefore, we mayadd the equation wjR' ' wjR! $ 0. The boundary conditions(40)–(42) are further simplified by assuming that/a ! / $ /! /c $ 1

2 Egap.Because the boundary conditions for the charge carrier

number densities (41) and (42) are independent of theapplied voltage, they have been called locked density bound-ary conditions, and their validity in polymer-fullerene solarcell device models has recently been questioned.33 Wewould like to emphasize that the three-dimensional chargetransport/recombination model described in Secs. III A andIII B may be completed with any type of boundary conditionfor the metal-insulator contacts.34

D. Post-processing and efficiency factors

From the solution %w; n; p& of the semiconductor equa-tions (17)–(19) with the boundary conditions (40)–(42), sev-eral quantities of practical interest may be computed. Fromthe electric current density,

j :$ !q%Jn ! Jp& "Am!2#; (43)

the electric current through the electrode boundaries may becomputed by integration

I6 :$!

R6

j ) n ds "A#; (44)

where n denotes the outward unit normal vector on @X. Thecurrents through the cathode and anode are of equal magni-tude with opposite signs. We have I' < 0 under illuminationat short circuit (Va $ 0), indicating that more electrons thanholes are flowing outwards through the cathode. The current

I' increases with the applied voltage Va, and it changes signat the open circuit voltage Voc "V#. For the simulation of thecurrent-voltage characteristic of a solar cell device, wedivide the current through the cathode, I'%Va&, by the areaof the primitive cell X0 used in the optics simulation(Sec. II A), i.e., we plot the current per unit area,

j%Va& :$ I'%Va&jX0j

"Am!2#; (45)

vs. the applied voltage Va (Fig. 6). The electric power perunit area is then given by

P%Va& :$ !Vaj%Va& "Wm!2#; (46)

which is positive for 0 < Va < Voc, under illumination. To-gether with the short circuit current Jsc :$ !j%0& "Am!2#, thefill factor FF and the energy conversion efficiency g of thesolar cell device can be computed

FF :$ Pout

JscVoc; g :$ Pout

Pin; Pout :$ maxVa P%Va&; (47)

where the input power per unit area, Pin, was defined in Eq.(16). The energy conversion efficiency g is to be maximizedvia shape optimization (Sec. IV).

Other interesting quantities obtained from post-processing include the exciton dissociation and charge car-rier collection efficiencies, which are defined by

gdiss :$ hPdissGihGi

; gcoll :$ hUihPdissGi

: (48)

The angled brackets h ) i in Eq. (48) denote spatial averagesover the domain X occupied by the polymer-fullerene blend.

We validate our charge transport simulation by reproduc-ing the results obtained by Koster et al.1 for a flat PPV:PCBMlayer of 120 nm thickness, with a constant exciton generationrate density. Our simulation results are stated in Table II—they are in good agreement with these earlier results, whichwere also verified by experimental measurements.1

FIG. 6. Current-voltage characteristics of the flat benchmark cell (dashedline) and of the double ridge pattern with maximum energy convergence ef-ficiency (solid line) (Sec. IV), obtained from the electro-photonic simulation(Secs. II and III). Current density j vs. applied voltage Va. The short circuitcurrent Jsc, open circuit voltage Voc, and fill factor FF (Sec. III D) for thetwo solar cell devices are given in the legend.

054502-7 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

IV. SHAPE OPTIMIZATION

With the electro-photonic simulation described in Secs. IIand III, we are now able to compute the energy conversion effi-ciency of solar cell devices under realistic illumination (47),given the periodic structure of the material layers and certainoptical and electrical material properties (Fig. 2, Table I). Wecombine this multiphysics simulation with the simulatedannealing metaheuristic (Appendix C) to find the cell geometrywhich maximizes the energy conversion efficiency for givenmaterial properties. This optimization aims at identifying pho-tonic crystal structures which result in solar cell devices with ahigher energy conversion efficiency than flat cells.

A. Results for flat cells and flat benchmark

We begin with a one-dimensional parameter space, whereonly the P3HT:PCBM layer thickness h3 is variable (Fig. 1).The other layer thicknesses are fixed to %h1; h2; h4&$ %178; 50; 0& nm. In Fig. 7, we plot the light absorption andcharge carrier collection efficiencies gabs (16), gdiss (48) at themaximum power point, as well as the energy conversion effi-ciency g (47) vs. h3. While more light is absorbed in the photo-active material as its volume increases, the loss of chargecarriers due to recombination increases, too. The optimal activelayer thickness is found at h3 $ 77 nm %g $ 00393&. Similar

values for the optimal P3HT:PCBM layer thickness in flat cellshave been reported before.35,36

Next, we consider a two-dimensional parameter space,where both the P3HT:PCBM and the nc-ZnO layer thicknessesh3; h4 are variable. The nc-ZnO material is non-absorbing inthe range of wavelengths considered here (Fig. 2) and thus actsas an optical spacer. Fig. 8 illustrates how the simulated anneal-ing metaheuristic automatically focuses in the “most inter-esting” region of the parameter space. The optimal parameterpair is found at %h3; h4& $ %53; 27& nm %g $ 0:0400&. We alsonote from Fig. 8 that the energy conversion efficiency is mainlygoverned by the sum h3 ' h4, an observation previously madealso for the short circuit current.37

Patterned cells shall be compared with the best possibleflat cell for this solar cell device model (Secs. II and III) andgiven material properties (Fig. 2, Table I). Therefore, wenow vary all four layer thicknesses, h1;…; h4 (Fig. 1) andshow the results in Fig. 9. The gray dots correspond to pointsin the parameter space generated by the simulated annealingmetaheuristic, shown together with the values of the energyconversion efficiency g. We find the maximum g $ 0:0433at %h1; h2; h3; h4& $ %78; 39; 51; 32& nm. This is our flatbenchmark cell.

B. Ridge pattern in the back electrode

We consider patterned solar cell devices with a planarfront electrode (flat glass, ITO, and Poly(3,4-ethylenedioxy-thiophene) poly(styrenesulfonate) (PEDOT:PSS) layers), anda periodic ridge pattern imprinted into the P3HT:PCBM bulkheterojunction material. The space between the ridges isfilled with nc-ZnO, and a flat aluminum layer closes the de-vice. For one particular pattern of this kind, the electric fieldlines and the net charge carrier generation rate at maximumpower, obtained from the electro-photonic simulation, are

TABLE II. Simulation results for a flat PPV:PCBM layer of 120 nm thick-ness, with constant exciton generation rate density.1 Applied voltage Va andcurrent per unit area j (45) at short circuit, maximum power, and open circuit

(Sec. III D). Also reported are the values of the dissociation efficiency gdiss

and of the recombination loss, which is computed from the collection effi-

ciency gcoll (48).

Va"V# !j%Va&"Am!2# gdiss"%# 1! gcoll[%]

Short circuit 0 28.8 60.2 6.37

Maximum power 0.658 19.5 50.4 23.7

Open circuit 0.844 0 46.5 97.5

FIG. 7. Simulation results for a flat layered solar cell device with variablethickness of the P3HT:PCBM layer, h3 (Fig. 1). The other layer thicknessesare fixed to %h1; h2; h4& $ %178; 50; 0& nm. Light absorption and charge car-rier collection efficiencies gabs (16), gdiss (48) at the maximum power point,as well as energy conversion efficiency g (47) vs. h3. Optimal thicknessfound at h3 $ 77 nm %g $ 00393&.

FIG. 8. Points in the two-dimensional parameter space (variableP3HT:PCBM and nc-ZnO layer thicknesses, h3; h4 (Fig. 1)) generated bythe simulated annealing metaheuristic. Other layer thicknesses fixed to%h1; h2& $ %178; 50& nm. Gray values indicate energy conversion efficiencyg (47) of the corresponding solar cell device. Optimal layer thicknessesfound at %h3; h4& $ %53; 27& nm %g $ 00400&.

054502-8 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

shown in Fig. 10. While these patterned solar cell devicesmay absorb more light than the flat cell with the sameamount of P3HT:PCBM bulk heterojunction material,2 theyalso suffer from a high recombination loss due to a chargecarrier sink (U < 0) in the photoactive layer. In our shapeoptimization efforts, we have not found any pattern of thiskind with a higher energy conversion efficiency than the flatbenchmark cell.

C. Ridge pattern in both front and back electrodes

Instead, we propose the pattern outlined in Fig. 3, whereridges of height h1 are etched into both the glass front layer

and the aluminum back layer. The remaining layers withthicknesses h2;…; h5 are coated conformally in-between.The widths of the two ridges, w1;w2, are also variable. Sucha pattern should offer the possibility of increased lightabsorption in the photoactive material due to a photonic crys-tal effect,14 while at the same time providing a fairly uniformtravel distance for the charge carriers.

We use the simulated annealing metaheuristic on the7-dimensional parameter space, and we plot the material vol-umes per unit area as black pluses in Fig. 9, together with thecorresponding value of the energy conversion efficiency g(47). It turns out that the optimal pattern allows for aboutthree times more P3HT:PCBM material than the flat bench-mark cell. Together with a much smaller volume of the frontlayers (ITO and PEDOT:PSS) the pattern increases the lightabsorption efficiency in the photoactive material by 14%compared to the flat benchmark cell (Fig. 4). The energyconversion efficiency g is improved by 15%. Optimalparameter values are given by %w1;w2; h1; h2; h3; h4; h5&$ %0; 0; 168; 4; 0; 28; 4& nm. This means pure ITO and nc-ZnO ridges of height 168 nm and width 8 nm on the frontand back, respectively, with a P3HT:PCBM layer of 28 nmthickness in-between (no PEDOT:PSS). The electric fieldlines in P3HT:PCBM for this pattern are shown in Fig. 11.Comb structures of this kind have also been found in blendmorphology optimization.38

D. Potential for further improvement

The energy conversion efficiency g of an organic bulkheterojunction solar cell device may be written as a productof five factors

g $ gabsggengdissgcollgout; (49)

where the exciton generation and output efficiencies aredefined by

FIG. 10. Electric field lines (solid) and labeled contour lines (dotted) of thenet charge carrier generation rate density U "1027 m!3s!1# at maximumpower (obtained from the electro-photonic simulation), for a solar cell de-vice with a ridge pattern imprinted into the back side of the P3HT:PCBMbulk heterojunction material and filled with nc-ZnO (Sec. IV B). The unit oflength is "nm#. This solar cell device suffers from a high recombination lossdue to a charge carrier sink (U < 0) in the photoactive layer.

FIG. 9. Gray dots: energy conversion efficiency g (47) vs. flat layer thicknesses h1;…; h4 (Fig. 1). Optimal values (flat benchmark): %h1; h2; h3; h4& $%78; 39; 51; 32& nm %g $ 00433& (Sec. IV A). Black pluses: g vs. volume per unit area of each material in a solar cell device with a double ridge pattern(Fig. 3). Maximum g $ 0:0497 found at %w1;w2; h1; h2; h3; h4; h5& $ %0; 0; 168; 4; 0; 28; 4& nm (Sec. IV C).

054502-9 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

ggen :$qVmp

jXjjX0j hGi

Pabs; gout :$ Pout

qVmpjXjjX0j hUi

: (50)

In Eq. (50), Vmp "V# denotes the applied voltage at the maxi-mum power point. ggen and gout relate the exciton generationrate to the power of absorbed light and the maximum outputpower to the charge carrier collection rate, respectively.Computed efficiency factors for both the benchmark flat cell(Sec. IV A) and the best patterned cell (Sec. IV C) are shownin Table III. The main energy loss mechanisms remaining af-ter shape optimization are the light absorption (loss due toreflection, transmission, and absorption in non-active materi-als) and the exciton generation from absorbed photons (lossdue to excess photon energy beyond the threshold requiredto produce an exciton): these two loss mechanisms alonelimit the efficiency of our solar cell device at gabsggen ’ 6 %.The absorption in the photoactive material within the win-dow of wavelengths 400 nm - k - 659 nm is already high

(Fig. 4); a substantial increase of the absorption efficiencygabs can therefore be achieved by using materials whichabsorb at higher wavelengths. Materials with long and thinabsorption tails seem particularly promising, since the pho-tonic crystal structure may lead to large gains in absorptionespecially in the tail region. Furthermore, it is necessary tomake use of effects which generate multiple excitons from asingle photon,12 in order to increase the energy conversionefficiency of exciton generation from absorbed photons, ggen.

V. CONCLUSIONS

Using a combined optics and charge transport simulationwe have computed the energy conversion efficiency for givensolar cell geometries and material parameters, under one-sunillumination. We have combined this simulation with thesimulated annealing metaheuristic to identify optimal geome-try parameters. A periodic pattern consisting of ridges of ITOand nc-ZnO, with a thin layer of P3HT:PCBM coated confor-mally in-between, improves the energy conversion efficiencyby 15% compared to the best possible flat cell for the givenmaterial parameters (also found by simulated annealing). Fur-ther investigation showed that after shape optimization, themain energy loss occurs in the light absorption and in the exci-ton generation from absorbed photons. These two loss mecha-nisms already limit the energy conversion efficiency of oursolar cell devices at .6 %.

ACKNOWLEDGMENTS

The authors would like to thank John R. Tumbleston(now at North Carolina State University) and Rene Lopez ofthe Department of Physics and Astronomy at the Universityof North Carolina at Chapel Hill for optical measurements13

and fruitful discussions throughout the project. This researchwas funded by the National Science Foundation within theSolar Energy Initiative (DMR 0934433).

APPENDIX A: OPTICS SIMULATION

The rigorous coupled wave analysis (Sec. II) allows fora mesh-free simulation of the light scattering and absorptionin the solar cell device. We restrict the reciprocal lattice K0

to the N > 0 shortest lattice vectors; then the generalizedeigenvalue problem (6) is of size 2N, and thus the series (5)becomes a finite sum. If the complex relative permittivity !r

FIG. 11. Electric field lines in P3HT:PCBM for the optimal double ridgepattern (Sec. IV C). The unit of length is "nm#.

TABLE III. Efficiency factors computed for the flat benchmark cell (Sec.IV A) and for the best pattern (Sec. IV C). The main energy loss mechanismsremaining after shape optimization are the light absorption and the genera-

tion of excitons from absorbed photons.

Flat benchmark Best pattern

gabs (16) 0.288 0.329

ggen (50) 0.206 0.200

gdiss (48) 0.839 0.880

gcoll (48) 0.929 0.963

gout (50) 0.935 0.891

g (47) 0.0433 0.0497

054502-10 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

does not vary in the horizontal directions, such as in flat lay-ered devices (Fig. 1), then N$ 1 (K0 $ f0g) is sufficient, andthe problem becomes one-dimensional. For N > 1, we com-pute the layer values of !r;G;G0 (4) analytically, since for apattern such as the one illustrated in Fig. 3, !r is piecewiseconstant in (x, y)-directions, in each layer.

We have used LAPACK39 to do the matrix computations

described in Secs. II A and II B. The computational cost ofthe QZ algorithm used for solving the generalized eigenvalueproblem (6) and of the LU decompositions used to solve thelinear systems in the transfer matrix method (for 10 differentlinear polarization angles h 2 f0/ ; 10

/;…; 90

/g (12)) isO%N3& floating point operations, as N !1. These problemsneed to be solved for a large number of wavelengths in orderto approximate the integral in Eq. (13). We have used N$ 7for the ridge patterns illustrated in Fig. 3. The exciton gener-ation rate (13) needs to be evaluated at each point in thecomputational mesh used for the charge transport simulation(Appendix B); the trapezoid rule was used for the numericalapproximation of the integral .

APPENDIX B: CHARGE TRANSPORT SIMULATION

1. Iterative scheme

Iteration is applied to transform the nonlinear system ofthree PDEs F(u)$ 0 (23)–(25) into a sequence of linear par-tial differential equations. A Gauss-Seidel-type iterationleads to a sequence of scalar nonlinear PDEs. The twocharge transport equations (24) and (25) become linear upondecoupling, and a single Newton step is used to solve (23) ineach iteration.40 Thus, we solve in each step the followingsequence of three linear, second-order, elliptic PDEs for u0,given the current iterate u

div%k2Dru0& ' A%u; v;w&u0 $ f %u; v;w&; (B1)

div%Dneu0rv0& ' B%u0&v0w $ g%u0&; (B2)

div%Dpe!u0rw0& ' B%u0&v0w0 $ g%u0&; (B3)

where the coefficients and source terms are given by

A%u; v;w& $ !veu ! we!u; (B4)

f %u; v;w& $ veu ! we!u ' A%u; v;w&u; (B5)

B%u0& $ !(

1! Pdiss%Vtjru0j; a; kf &)cNint; (B6)

g%u0& $ B%u0& ! Pdiss%Vtjru0j; a; kf &G=Nint; (B7)

with the dissociation probability Pdiss (35). Alternatively, aNewton-type iteration can be applied to F(u)$ 0, which lin-earizes F (23)–(25) about the current iterate u so that a sys-tem of linear PDEs needs to be solved in each step. Whilethe Newton-type iteration is expected to converge faster thanthe Gauss-Seidel-type iteration (B1)–(B3), it may also faildue to a bad starting point or inexact linearization of F, forexample. Since we need to compute solutions for several dif-ferent boundary values (to find the maximum power point)and also for various domain geometries (shape optimization),

we have opted for the more robust Gauss-Seidel-typemethod. Iteration (B1)–(B3) is run until the 2-norm of thenonlinear residual, kF%u&k2, falls below a prescribedtolerance.

2. Discretization

As we have mentioned in Sec. III C, the computationaldomain X is bounded by the electrode boundaries R6 and bythe faces of the right prism X0 *R, where X0 ( R2 denotesthe primitive cell of the lattice K used in the optics simula-tion (Sec. II A). We have found it convenient to discretize apart of the full right prism first, and to mask out the meshcells which do not contain polymer-fullerene blend material(Fig. 12). These cells are used to store the boundary values.

The second-order differential operators in Eqs. (B1)–(B3) are of the form div%Dr/&, with the unknown function/ and with a variable coefficient D > 0. An approximationof the divergence at the cell centers using centered finite dif-ferences requires the values of the flux Dr/ at the cell faces.Assuming a uniform mesh of cubic cells with edge lengthh > 0, we use the following approximation of the flux in onespace dimension

%D/0&i'1=2 ’1

h

!xi'1

xi

1

D%x&dx

0

@

1

A!1

/i'1 ! /i

h; (B8)

where /i denote the values of the unknown function / at thecell centers xi. In higher space dimensions, we may simplydiscretize in the same way in each direction, because the dif-ferential operator separates in cartesian coordinates. The dis-cretization (B8) applied to Eqs. (B1)–(B3) leads to anexponential upwind scheme, which is commonly used tosolve the semiconductor equations.41

The discrete versions of (B1)–(B3) each give rise to asparse linear system (no more than 7 non-zero entries in eachmatrix row in 3D), which we solve iteratively using the

FIG. 12. Illustration of the computational mesh for a patterned solar cell de-vice (cf., Fig. 3). Mesh cells which do not contain polymer-fullerene blendmaterial are masked out (*); they are used to store the electrode boundaryvalues (Sec. III C).

054502-11 C. Kirsch and S. Mitran J. Appl. Phys. 112, 054502 (2012)

conjugate gradient method.42 We also employ a simple pre-conditioner given by the diagonal approximate inverse,43 toaccelerate convergence.

The evaluation of the residual F%u0& at the end of eachiteration step is straightforward, because the operatorsappearing in Eqs. (23)–(25) are discretized anyway duringthe iteration.

3. Dissociation probability

The dissociation probability Pdiss%Vtjru0j; a; kf & needsto be evaluated before step (B2) in the iteration, from the so-lution u0 of Eq. (B1). We use centered finite differences tocompute the local electric field strength from the cell-centervalues of u0. The integral (35) is transformed into an integralover the whole real line by writing x $ exp%y&; y 2 R. Then,the integrand exhibits a doubly-exponential decay asy! 61. Thus, we may use the trapezoid rule for theapproximation, which converges exponentially in this case.44

The dissociation rate (32) is evaluated using the fast-converging series in Eq. (33), by truncation.

4. Post-processing

We vary the applied voltage Va in a one-dimensionalsearch and use the false position method to find the open cir-cuit voltage Voc, as well as golden section search45 to deter-mine the maximum power Pout (47) without requiringderivatives of j (45).

APPENDIX C: SIMULATED ANNEALING

The simulated annealing metaheuristic3,4 is used in globaloptimization, such as for maximizing the power output perunit area of a solar cell device in our application, with a vari-able cell geometry. Simulated annealing is based on theMetropolis-Hastings (MH) algorithm,46,47 which is a Markovchain Monte Carlo method for obtaining random samplesfrom a (typically complicated) probability distribution.

For N > 0 geometry parameters (such as layer thick-nesses and ridge width (Fig. 3)), we consider a real-valued,N-dimensional random vector X, as well as an objectivefunction g : RN ! R which is to be minimized (here, thenegative of the maximum power per unit area, !Pout (47),for a solar cell geometry (“state”) described by realizationsx 2 RN of X). In our application, evaluating g(x) corre-sponds to running the electro-photonic simulation to find themaximum power point for a cell geometry described by x.We define the random variable Y:$ g(X) for which we pre-scribe a probability density function fT . The function fTdepends on the “temperature” T > 0 in the simulated anneal-ing metaheuristic. The probability density function of X isthen given by pT :$ fT ! g, and the MH algorithm is used toconstruct a sample sequence of a Markov chain with equilib-rium distribution pT . We assume a fixed T > 0 for themoment and omit the index T. The MH algorithm employsan arbitrary Markov chain with transition probabilityq%x0jx&; x; x0 2 RN . For every fixed x 2 RN , the functionq% ) jx& is a probability density function in the state space,which is called the proposal density. Typically, the proposal

density is chosen such that its location is the current state x,and there may be additional parameters involved, such as thevariance. These additional parameters may also change dur-ing the course of the MH algorithm, to reduce the mixingtime.48 The algorithm constructs a sample sequence of a newMarkov chain fXkgk2N0

with transition probability

p%x0jx& $ q%x0jx&min 1;p%x0&q%xjx0&p%x&q%x0jx&

* +; (C1)

by accepting proposal states x0 sampled from q% ) jx& with aprobability that depends on the ratio of probabilities of thestates x and x0. It is easily verified that p is indeed a station-ary distribution of the Markov chain fXkgk2N0

. Therefore,after a sufficiently long “burn-in” period, the MH algorithmgenerates samples from p. Introducing the temperature T > 0again, the typical choice for the probability density functionfT of Y in simulated annealing is proportional to the Boltz-mann factor,

fT%y& / exp%!bTy&) pT%x& / exp%!bTg%x&&; (C2)

with the thermal beta bT :$ %kBT&!1, where kB denotes theBoltzmann constant. Notice that the value of the normaliza-tion constant in Eq. (C2) is not important, because only ratiosof values of pT are used in Eq. (C1). This is a benefit of theMH algorithm, because the partition function is difficult tocompute in general. pT is thus interpreted as the probabilitydensity function of states of a system in thermodynamicequilibrium at temperature T, where the thermodynamicpotential is given by the objective function g. In simulatedannealing, the temperature T is gradually reduced accordingto a “cooling schedule” and the MH algorithm is used tosample pT for each value of T. Notice that for T > 0, pro-posal states x0 with g%x0& > g%x& are occasionally accepted,which should prevent the algorithm from getting stuck in alocal minimum of g. As T decreases, so does the expectationEpT "g%X&#, and therefore we expect to reach a global mini-mum of g as T ! 0.

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