Slow dynamics of spin pairs in random hyperfine field: Role of inequivalence of electrons and holes in organic magnetoresistance
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Slow dynamics of spin pairs in random hyperfine field: Role of inequivalence ofelectrons and holes in organic magnetoresistance
R. C. Roundy and M. E. Raikh
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112
In an external magnetic field B, the spins of the electron and hole will precess in effective fields b e + B and b h + B , where b e and b h are random hyperfine fields acting on the electron and hole,respectively. For sparse “soft” pairs the magnitudes of these effective fields coincide. The dynamics ofprecession for these pairs acquires a slow component, which leads to a slowing down of recombination.We study the effect of soft pairs on organic magnetoresistance, where slow recombination translatesinto blocking of the passage of current. It appears that when b e and b h have identical gaussiandistributions the contribution of soft pairs to the current does not depend on B . Amazingly, smallinequivalence in the rms values of b e and b h gives rise to a magnetic field response, and it becomesprogressively stronger as the inequivalence increases. We find the expression for this response byperforming the averaging over b e , b h analytically. Another source of magnetic field response in theregime when current is dominated by soft pairs is inequivalence of the g -factors of the pair partners.Our analytical calculation indicates that for this mechanism the response has an opposite sign. PACS numbers: 73.50.-h, 75.47.-m
I. INTRODUCTION
Due to complex structure of organic semiconductorsand their spatial inhomogeneity it is nearly impossibleto identify a unique scenario of current passage throughthem. In view of this, it is remarkable that sizablechange of current through a device based on organic semi-conductor takes place in weak external magnetic fields.This effect, called organic magnetoresistance (OMAR),seems to be robust, i.e. weakly sensitive to the deviceparameters. Although the first reports on the obser-vation of organic magnetoresistance (OMAR) appeareddecades ago , systematic experimental study of this ef-fect started relatively recently. (see also the reviewRef. 20).On the theory side, it is now commonly accepted thatthe origin of OMAR lies in random hyperfine fields cre-ated by nuclei surrounding the carriers (polarons). Morespecifically, the basic unit responsible for OMAR is a pairof sites hosting carriers (polarons); the spin state of thepair is described by the Hamiltonian b H = Ω · b S e + Ω · b S h . (1)Here b S e are b S h are the spin operators of the pair-partners(we will assume that they are electron and hole, respec-tively); Ω = B + b e and Ω = B + b h are the full fieldsacting on the spins. They represent the sums of exter-nal, B , and respective hyperfine fields, b e and b h . As wasfirst pointed out by Schulten and Wolynes , due to thelarge number of nuclei surrounding each pair-partner andtheir slow dynamics, b e and b h can be viewed as classicalrandom fields with gaussian distributions.In order to give rise to OMAR the HamiltonianEq. (1) is not sufficient. It should be complementedby some mechanism through which the pair-partners“know” about each other, so their motion is correlatedwithout direct interaction. The simplest example of such Ω Ω θ θ xz | Ω ,x || Ω ,x | B W b e B W b h FIG. 1: Preferential coordinate system used for analysis ofthe dynamics of the spin pair. Both fields Ω , Ω reside inthe xz -plane. The direction of the quantization axis, z , isfixed by the condition Ω ,x = − Ω ,x . a mechanism is spin-dependent recombination, i.e. therequirement that electron and hole can recombine only iftheir spins are in the singlet, S , state. Then the essence ofOMAR can be crudely understood as a redistribution ofportions of singlets and triplets upon increasing B . Thisredistribution affects the net recombination rate. Clearly,the characteristic B for this redistribution is ∼ b e , b h .Naturally, the specific relation between the current andrecombination rate involves also the rate at which thepairs are created. It is important, though, that the latterprocess is not spin-selective.Existing theories of OMAR can be divided into twogroups which we will call “steady-state” and “dynami-cal”. The theories of the first group appeared earlier.In a nutshell (see Ref. 18 for details), in these theo-ries the right-hand-side of the equation of motion for thedensity matrix i ˙ ρ = [ b H, ρ ] with Hamiltonian Eq. (1)is complemented with “source” and spin-selective “sink”terms. After that, ˙ ρ is set to zero. In Refs. 15 cur-rent is expressed via the steady-state ρ and subsequentlyaveraged numerically over realizations of hyperfine fields.The “steady-state” approach applies when the pairdoes not perform many beatings between S and T dur-ing its lifetime, since the beating dynamics is excludedby setting ˙ ρ = 0.This beating dynamics has been incorporated into theOMAR theory Ref. 22, which appeared last year. Thistheory relies on decades old findings in the field of dy-namic spin-chemistry . Below we briefly summarizethese findings.If an isolated pair is initially in S , it was shown in Ref.4 that the averaged probability to find it in T after time t is given either by the function p ST ( t ) = 12 (cid:16) − e − b e t e − b h t (cid:17) , (2)for strong fields B ≫ b e , b h , or by p ST ( t ) = 34 (cid:26) − (cid:20) (cid:16) e − b e t − b e t e − b e t (cid:17)(cid:21) × (cid:20) (cid:16) e − b h t − b h t e − b h t (cid:17)(cid:21)(cid:27) , (3)for B ≪ b e , b h . Here b e , b h are the rms hyperfine fields forelectron and hole. Naturally, the probability approaches3 / B and 1 / B .In the theory of Ref. 22 the B -dependent dynamics de-scribed by Eqs. (2), (3) translates into the B -dependentresistance (OMAR) on the basis of the following reason-ing. The dynamics p ST ( t ) leads to prolongation of therecombination time (hopping time, τ h , in the language ofRef. 22). This prolongation is quantified by1 τ h → τ h Z dt (1 − p ST ( t )) e − t/τ h . (4)The meaning of Eq. (4) is that a pair should stay in S in order for a hop to take place. Prolongation of hoppingtime leads to a B -dependent increase of the resistance.The authors of Ref. 22 evaluated p ST ( t ) for arbitrary B ,while in calculation of OMAR they assumed that barehopping times, τ h , have an exponentially broad distribu-tion.Both theories Refs. 18, 22 take as a starting point apair with the Hamiltonian Eq. (1) describing its spinstates and preferential recombination (hopping) from S .The dynamics of this seemingly simple entity, which iscrucial for OMAR, possesses some nontrivial regimes.Uncovering these regimes is a central goal for the presentpaper. The other goal is to demonstrate that nontrivialdynamics can manifest itself in OMAR.To underline that the spin dynamics of two carriers innon-collinear magnetic fields which can recombine onlyfrom S can be highly nontrivial, we note that separa-tion of this dynamics into S - T “beating” stage followed by instantaneous hopping after time τ h , as in theory 22,is not always possible. It is quite nontrivial that spin-selective recombination of carriers can exert a feedback on the spin dynamics. As an illustration of this deli-cate issue we invoke the example of cooperative photonemission discovered by R. H. Dicke . In the Dicke ef-fect one superradiant state of a group of emitters havinga very short lifetime automatically implies that all theremaining states are subradiant and have anomalouslylong radiation times. Below we demonstrate that a sim-ilar situation is realized in dynamics of two spins whenrecombination from S is very fast. We will see that theremaining 3 modes of the collective spin motion becomevery “slow”.Our analysis reveals the exceptional role of the “soft”pairs, which are sparse configurations of b e , b h for whichfull fields Ω , Ω have the same magnitude.The paper is organized as follows. In Sect. II we castthe eigenmodes of the Hamiltonian Eq. (1) in a conve-nient notation. In Sect. III we include recombination andstudy its effect on the eigenmodes. The consequences ofnontrivial dynamics for OMAR are considered in Sects.IV and V, where we perform averaging over realizationsof hyperfine fields. We establish that inequivalence of rmshyperfine fields for electrons and holes has a dramatic ef-fect on OMAR, when it is governed by soft pairs. Sect.VI concludes the paper. II. DYNAMICS OF A PAIR IN THE PRESENCEOF RECOMBINATIONA. Isolated pair
We start with reviewing the dynamics of a pair of spinsin the absence of recombination. Obviously, this dynam-ics does not depend on the choice of the quantizationaxis. However, since we plan to include recombination,the choice of the quantization axis, z , illustrated in Fig.1 appears to be preferential. The axis is chosen to liein the plane containing the vectors Ω , Ω . Moreover,the orientation of the z -axis is fixed by the condition Ω x = − Ω x . Then the angles, θ , θ , between Ω , Ω and the z -axis are given bytan θ = | Ω × Ω | Ω + Ω · Ω , tan θ = | Ω × Ω | Ω + Ω · Ω (5)With this choice, the Schr¨odinger equation for the am-plitudes of S , T , T + , and T − reduces to the system i ∂T + ∂t = Σ z T + − √ x S, (6) i ∂S∂t = ∆ z T − √ x T + + 1 √ x T − , (7) i ∂T ∂t = ∆ z S, (8) i ∂T − ∂t = − Σ z T − + 1 √ x S, (9)where Σ z , ∆ z , and ∆ x are defined asΣ z = Ω z + Ω z , (10)∆ z = Ω z − Ω z , (11)∆ x = Ω x − Ω x . (12)The advantage of our choice of the quantization axisshows in the fact that the state T is coupled exclusivelyto S . Since recombination is allowed only from S , thiswill simplify the subsequent analysis of the recombina-tion dynamics.The eigenvalues, λ i , of the system Eqs. (6)-(9) satisfythe quartic equation λ i ( λ i − Σ z ) − λ i (∆ z + ∆ x ) + ∆ z Σ z = 0 . (13)We will enumerate these eigenvalues according to theconvention λ = − λ and λ = − λ . To find the absolutevalues λ , λ one does not have to solve Eq. (13), since itis obvious that for non-interacting spins the eigenvaluesare the sums and the differences of individual Zeemanenergies λ = (cid:18) | Ω | + | Ω | (cid:19) , λ = (cid:18) | Ω | − | Ω | (cid:19) . (14)Naturally, λ , λ do not depend on the choice of thequantization axis. At the same time, the coefficients inEq. (13) do depend on this choice. To trace how thedependence on the quantization axis disappears in theroots of Eq. (13), one should use the following identitiesΣ z + ∆ z + ∆ x = | Ω | + | Ω | , (15)Σ z ∆ z = (cid:18) | Ω | − | Ω | (cid:19) . (16)In terms of the angles θ and θ , Fig. 1, the corre-sponding eigenvectors can be expressed as T + ST T − = cos θ cos θ − √ sin θ + θ √ sin θ − θ − sin θ sin θ , − sin θ sin θ − √ sin θ + θ − √ sin θ − θ cos θ cos θ , cos θ sin θ √ cos θ + θ √ cos θ − θ sin θ cos θ , − sin θ cos θ − √ cos θ + θ √ cos θ − θ − cos θ sin θ , (17)where the first two correspond to λ , while the last twocorrespond to λ , , respectively.The form Eq. (17) allows us to make the following ob-servation. When the full magnetic fields acting on spinsincidentally coincide, we have | Ω | = | Ω | . Then it fol-lows from Eq. (14) that λ = λ = 0, so that the two cor-responding eigenstates become degenerate. Under this condition we also have θ = θ . Then the first two eigen-vectors Eq. (17) have zeros in the rows corresponding to T . Concerning the other two eigenvectors, due to theirdegeneracy, their sum and difference are also eigenvec-tors. The difference has a zero in the row correspondingto T , while the sum consists of the T component, exclu-sively. Then we conclude that for realizations of hyper-fine field for which | Ω | = | Ω | the state T is completelydecoupled from the other three states. This fact has im-portant implications for recombination dynamics, as wewill see below.Including recombination requires the analysis of thefull equation for the density matrix i ˙ ρ = [ b H, ρ ] − i τ { ρ, | S i h S |} , (18)where τ is the recombination time. The form of the sec-ond term ensures that recombination takes place onlyfrom S . The matrix corresponding to Eq. (18) is 16 × λ i − λ ∗ j , where λ i and λ j satisfy the equation λ i (cid:18) λ i + iτ (cid:19) ( λ i − Σ z ) − λ i (∆ z + ∆ x ) + ∆ z Σ z = 0 . (19)The latter equation expresses the condition that λ i arethe eigenvalues of non-hermitian operator b H − iτ | S i h S | .In the limit τ → ∞ this equation reduces to Eq. (14).The dynamics of recombination is governed by the imagi-nary parts of the roots of Eq. (19), i.e. decay is describedby the exponents exp [ − (Im λ i + Im λ j ) t ]. Less trivialis that finite τ can strongly affect the real parts of λ i .Physically, the dependence of Re λ i on τ describes the back-action of recombination on the dynamics of beatingbetween different eigenstates. In the following two sub-sections this effect will be analyzed in detail in the twolimiting cases. B. Slow Recombination
Consider the limit 1 /τ ≪ λ i . In this limit recombina-tion amounts to the small corrections to the bare valuesof λ i given by Eq. (14). This allows one to set λ i equalto their bare values in all terms in Eq. (19) containing1 /τ , and search for solution in the form λ i + δλ i . Thenone gets the following expression for the correction δλ i δλ i = − iτ λ i ( λ i − Σ z ) λ i − λ i (Σ z + ∆ z + ∆ x ) = − i τ λ i ( λ i − Σ z ) λ i − Σ z ∆ z . (20)In the last identity we have used the fact that λ i satisfythe equation Eq. (14). The above expression can begreatly simplified with the help of the relations Eq. (15). λ − λ λ − λ a. λ − λ λ − λ a. δλ δλ λ λ λ − λ b. δλ δλ λ λ λ − λ b. T + ST T − c. T + ST T − c. FIG. 2: (Color online). (a) Slow-recombination regime,Ω , ≪ /τ . Horizontal lines represent the energy levels Eq.( ) of a pair in the absence of recombination. Recombina-tion from S causes the broadening of the levels Eqs. ( ),which, for a typical pair, is of the same order for all levels. (b)Slow-recombination regime. For soft pairs , | Ω | ≈ | Ω | , re-combination results in splitting Eq. ( ) of the widths of thelevels λ , rather than their positions. (c) Fast-recombinationregime, Ω , ≫ /τ . The eigenstates S , T , T + , and T − arewell-defined. Recombination causes strong broadening, 1 /τ ,of the level S , and weak broadening ∼ Ω , τ of the other threelevels. One has δλ , = − i τ (cid:18) − Ω · Ω | Ω || Ω | (cid:19) , (21) δλ , = − i τ (cid:18) Ω · Ω | Ω || Ω | (cid:19) . (22)The above result suggests that for generic mutual ori-entations of Ω and Ω all modes of a pair decay withcharacteristic time ∼ τ . At the same time, for parallelorientations of Ω , Ω the modes λ , have anomalouslylong lifetime. This long lifetime has its origin in the factthat for Ω k Ω , the states T + and T − , which are orthog-onal to S , are the eigenstates of the Hamiltonian Eq. (1).Formally this can be seen from the general expression Eq.(17) for the eigenvectors upon setting θ = θ = 0. Sim-ilarly, for Ω and Ω being antiparallel, one can checkfrom Eq. (17) that for θ = π − θ , the eigenstates λ , λ have no S component, so they are long-lived. Note thatthe existence of long lifetimes for parallel and antiparallelconfigurations of Ω , Ω is at the core of the “blockingmechanism” of OMAR proposed in Ref. 15.
1. Soft pairs
As was pointed out in the Introduction, recombinationalso has a pronounced effect on the spin dynamics forsparse configurations for which | Ω | ≈ | Ω | . Indeed, forthese configurations, the values λ and λ are anoma-lously small. Then the basic condition, 1 /τ ≪ λ i , underwhich Eq. (21) was derived, is not satisfied. We dub suchrealizations as soft pairs. For soft pairs the expressionsfor δλ , δλ remain valid, but the eigenvalues λ , λ getstrongly modified due to finite recombination time, τ .Although for soft pairs the terms ∝ /τ in Eq. (19)cannot be treated as a perturbation, a different simplifi-cation becomes possible in this case. We can neglect λ i compared to Σ z in the first term and ∆ z compared to ∆ x in the second term. The first simplification is justified,since the typical value of Σ z is ∼ | Ω | ≈ | Ω | and is muchbigger than both 1 /τ and ( | Ω | − | Ω | ). Concerning thesecond simplification, the smallness of ( | Ω | − | Ω | ) au-tomatically implies that ∆ z given by Eq. (10) is small.With the above simplifications the eigenvalues λ , sat-isfy the quadratic equation(Σ z + ∆ x ) λ i + iτ Σ z λ i − ∆ z Σ z = 0 . (23)Already from the form of Eq. (23) one can make a sur-prising observation that, even with finite 1 /τ , one of theroots is identically zero when ∆ z = 0, i.e. when | Ω | and | Ω | are exactly equal to each other. This suggests thata pair in the state corresponding to this root will neverrecombine. For a small but finite difference ( | Ω | − | Ω | )the recombination will eventually take place but only af-ter time much longer than τ . Indeed, for the generic case, | Ω | ∼ | Ω | ), we have from Eq. (23) λ , = − i τ h Λ ± p Λ − z τ i , (24)where the dimensionless parameter Λ is defined asΛ = Σ z Σ z + ∆ x . (25)Even when | Ω | and | Ω | are close, a typical value of pa-rameter Λ is ∼
1. Then Eq. (24) suggests that anoma-lously long-living mode exists in the domain ∆ z . /τ where its lifetime is ∼ / ∆ z τ . Note that the lifetime be-comes longer with a decrease of the recombination time.As the difference | Ω | − | Ω | increases, the product∆ z τ becomes big and the expression under the squareroot in Eq. (24) becomes negative. Then the lifetimes ofof both states corresponding to λ and λ become equalto τ / Λ. Note that, at the same time, the splitting of thereal parts of λ and λ becomes ∼ ∆ z τ , which is muchbigger than | Ω | − | Ω | .The above effect can be interpreted as a repulsion of the eigenvalues caused by recombination . A moreprominent analogy can be found in optics . The signs+ and − in Eq. (24) can be related to the superradiantand subradiant modes of two identical emitters. The roleof τ in this case is played by their radiative lifetime.Both effects illustrate the back-action of recombina-tion on the dynamics of the pair when the spin levels ofpair-partners are nearly degenerate. To track an analogyto this effect one can refer to Refs. 24 and 25, whereEq. (24) appeared in connection to resonant tunnelingthrough a pair of nearly degenerate levels, while the roleof 1 /τ was played by the level width with respect to es-cape into the leads.For our choice of the quantization axis the long-livingstate corresponds to T . For completeness we rewrite theparameter ∆ z , which enters Eq. (24), in the coordinate-independent form∆ z = ( | Ω | − | Ω | ) | Ω + Ω | . (26)To establish coordinate-independent form of parameterΛ we need the combinations Σ z and Σ z + ∆ x , which aregiven by Σ z = | Ω + Ω | , (27)Σ z + ∆ x = | Ω | + | Ω | − (cid:0) | Ω | − | Ω | (cid:1) | Ω + Ω | , (28)so that Λ can be cast into the formΛ = | Ω + Ω | | Ω + Ω | + 4 | Ω × Ω | . (29)The consequences of “trapping” described by Eq. (24)for OMAR will be considered in Sections IV and V. Inthe subsequent subsection we will see that the similarphysics, namely, the emergence of slow modes due to fast recombination persists also in the domain | Ω , | τ ≪ C. Fast Recombination
In the opposite limit, τ ≪ | Ω , | − , the bracket ( λ i + iτ )in Eq. (19) is big. This suggests that three zero-ordereigenvalues are λ i = 0 , ± Σ z . (30)In the same order, the fourth eigenvalue is − iτ . Concern-ing the eigenvectors, in the zeroth order they are simply S, T + , T − , and T . This follows from the equation i ˙ S + iτ S = ∆ z T − √ T + + 1 √ T − . (31)Taking τ to zero means that in the zeroth order S = 0.Then three other equations in the system Eq. (6) getdecoupled.In the first order, the eigenvalues Eq. (30) acquireimaginary parts δλ i = − iτ (cid:18) λ i (∆ z + ∆ x ) − ∆ z Σ z λ i − Σ z (cid:19) . (32)With the help of Eqs. (27) and (29) these imaginaryparts can be simplified to δλ T = − iτ ∆ z = − iτ (Ω − Ω ) | Ω + Ω | , (33) δλ T + = δλ T − = − iτ ∆ x − iτ | Ω × Ω | | Ω + Ω | . (34)We see that for a generic situation | Ω | ∼ | Ω | the life-time of the modes T , T + , and T − are ∼ / | Ω , | τ , i.e. in the regime of fast recombination it is much longer than τ . This is a consequence of effective decoupling of T , T + ,and T − from S in this regime. We also observe from Eq.(33) that there is additional prolongation of lifetime forthe mode T if the pair is soft. Eq. (33) also suggeststhat lifetimes of the states T + , T − are anomalously longwhen Ω and Ω are collinear. This expresses the obvi-ous fact that, for collinear effective fields acting on thepair-partners, T + and T − are the eigenstates no matterwhether recombination is present or not.Once the eigenvalues and eigenvectors of a pair in thepresence of recombination are established, the next ques-tion crucial for transport through the pair is: Supposethat initial state is a random superposition of S , T , T + ,and T − , what is the average (over the coefficients of su-perposition) waiting time for this state to recombine?Naturally, the answer to this question does not dependon the actual choice of the orthonormal basis. We ad-dress this question in the next section. III. RECOMBINATION TIME FROM ARANDOM INITIAL STATEA. Soft pair in a slow recombination regime
To illustrate the peculiarity of the question posedabove, we start from an instructive particular case ofsoft pair in a slow recombination regime. We defined asoft pair as a pair for which the condition ( | Ω |−| Ω | ) ≪| Ω , | is met. However, in the slow recombination regime,the combination ( | Ω |−| Ω | ) τ can be either big or small.In both cases there is a strong separation between the ab-solute values of λ , and λ , . It can be seen from Eq.(24) that in the limit ( | Ω | − | Ω | ) τ ≫
1, the recombina-tion times for states which correspond to λ and λ aregiven by t (3) R = 2 τ Λ and t (4) R = 2 τ Λ , (35)while in the opposite limit, ( | Ω | − | Ω | ) τ ≪
1, we get t (3) R = 2 τ Λ and t (4) R = 1 τ ∆ z . (36)We see that the recombination time of λ is ∼ τ forboth limits, while the recombination time of λ crossesover from ∼ τ to ∼ / ∆ z τ as ( | Ω | − | Ω | ) τ decreases.Taking into account that for generic case | Ω | ∼ | Ω | therecombination times corresponding to λ , are ∼ τ , weconclude that for purely random initial conditions theaverage recombination time is either ∼ τ or it is of1 / ∆ z τ .The major complication for getting exact average re-combination time for a soft pair is that the exact eigen-states represent mixtures with weights governed by therecombination time. This follows from Eq. (24). In addi-tion, the eigenstates corresponding to λ , and λ are notorthogonal to each other. However, for a soft pair thesecomplications can be overcome. The reason is that, thereare two small parameters in the problem, 1 /τ | Ω , | , and( | Ω |− | Ω | ) / | Ω , | . The first parameter guarantees slowrecombination, while the second ensures that the pair issoft. The presence of these parameters allows us to eval-uate h t R i in the closed form using the general formula h t R i = 14 X i,j g ji ( g − ji ) ∗ i ( λ i − λ ∗ j ) , (37)where g ij = h v i | v j i is a matrix of inner products of eigen-vectors corresponding to complex eigenvalues λ i and λ j .The above formula becomes absolutely transparent whenthe eigenvectors are orthonormal. Then the matrix g ij reduces to the Kronecker symbol, δ ij , and h t R i simplifiesto h t R i = − X j λ j , (38)which expresses the fact that for random initial state theaverage recombination time is the evenly-weighted sumof recombination times from eigenstates.In the case of a soft pair and slow recombination oneshould use Eq. (37) to evaluate h t R i . What enablesthis evaluation is that, by virtue of small parameters,the eigenvectors corresponding to λ and λ are mutu-ally orthogonal (with accuracy 1 /τ | Ω , | ), and they areboth orthogonal to eigenvectors corresponding to λ and λ . Therefore, in evaluating Eq. (37), one has to dealonly with mutual non-orthogonality of two eigenvectors v and v . The straightforward calculation yields h t R i = τ Λ + 14∆ z τ − λ − λ , (39)where Im λ = Im λ are given by Eq. (21). It iseasy to see that in the limiting cases of large and small( | Ω | − | Ω | ) τ Eq. (39) reproduces Eqs. (35) and (36),respectively.While in the last two terms in Eq. (39) depend weaklyon the degree of “softness” of the pair, ∆ z ∝ ( | Ω |−| Ω | ),the second term exhibits unlimited growth with decreas-ing ∆ z . We emphasize the peculiarity of this situation.In conventional quantum mechanics, when the level sep-aration becomes smaller than their width, it should besimply replaced by the width. What makes Eq. (39)special is that the smaller is ∆ z the more the state T becomes isolated. There is direct analogy of this situ-ation with the Dicke effect , as was mentioned in theIntroduction. By virtue of this analogy, the state T as-sumes the role of the “subradiant” mode which accom-panies the formation of the superradiant mode. In theDicke effect the formation of superradiant and subradi-ant states occurs because the bare states are coupled viacontinuum. In our situation it is recombination that is re-sponsible for “isolation” of T . If the pair is not soft, thecalculation of the time h t R i in the slow-hopping regime LW I n- I n I n I + B W b B W b FIG. 3: (Color online). The simplest model of transportthrough a bipolar device in which the currents flow along inde-pendent chains. Electrons arrive at the recombination regionfrom the left, while the holes arrive from the right. Blobs en-close the sites from which electron and hole recombine. Oneof the blobs is enlarged to illustrate the spin precession of thepair partners in their respective fields Ω , Ω . For soft pairsthe magnitudes of Ω and Ω are close to each other. can be performed by simply using Eq. (38) and λ i givenby Eqs. (21), (24). This is because the smallness of1 /τ makes the eigenstates almost orthogonal. However,the Dicke physics becomes even more pronounced in thefast-recombination regime, as demonstrated in the nextsubsection. B. Recombination time in the fast recombinationregime
It might seem that under the condition of fast recom-bination | Ω , | τ ≪ ∼ τ , since spins practicallydo not precess during the time τ . The fact that recom-bination takes place only from S , while initial state is arandom mixture, already suggests that h t R i is longer than τ . This is because if the initial configuration is differentfrom S it must first cross over into S by spin precessionbefore it recombines. The characteristic time for the spinprecession is ∼ | Ω , | − ≫ τ . It turns out that the cross-ing time is actually much longer than | Ω , | − . Formally,this fact follows from Eqs. (33), (34) for δλ i , which areof the order of | Ω , | τ rather than | Ω , | . We can nowinterpret this result by identifying S with superradiantstate, while T , T + , and T − assume the roles of subra-diant states. The short lifetime of S isolates it from therest of the system. Quantitatively, the portion of S inthe other eigenvectors is ∼ | Ω , | τ .What is important for calculation of h t R i is thefact that eigenvectors are orthogonal (with accuracy ∼ / | Ω , | τ ) in the fast-recombination regime. This allowsone to replace the overlap integrals g ij in Eq. (37) by δ ij and use the Eq. (38) which immediately yields for h t R i I II IVIII t GS - T + + T T- FIG. 4: (Color online). I, II, III, and IV are possible variantsof the current cycle. For each variant the pair is initially cre-ated in one of four states. This is followed by time evolution,illustrated by blue double arrows, which mixes the states.Subsequently, the pair either recombines from S (brown ar-row) or dissociates. The processes of creation and dissocia-tion are indicated by white double arrows. The current is theinverse duration, t , of the cycle averaged over initial states,which we assume to have equal probabilities. The time, t , isgiven by Eqs. ( ), ( ), or ( ) depending on the recombi-nation regime. the result h t R i = − (cid:18) λ S + 1Im λ T + 1Im λ T + + 1Im λ T − (cid:19) (40)= 18 (cid:20) τ + 1 τ (cid:18) z + 4∆ x (cid:19)(cid:21) . (41)Substituting the coordinate-independent expressions for∆ x and ∆ z , we arrive at the final expression for recom-bination time, which is applicable within the entire fast-recombination regime h t R i = 18 " τ + 4 τ | Ω + Ω | ( | Ω | − | Ω | ) + | Ω + Ω | | Ω × Ω | ! . (42)As was already noticed in the previous section, recom-bination time diverges for two particular configurations:soft pairs with | Ω | = | Ω | and collinear Ω and Ω .Certainly this divergence will be cut off in the course ofcalculation of current through a pair to which we nowturn. IV. TRANSPORT MODEL
We adopt a transport model illustrated in Fig. 3. Forconcreteness we will discuss a bipolar device, so that the current is due to electron-hole recombination. As shownin Fig. 3, electrons arrive at the pair of sites (enlargedregions in Fig. 3) from the left, while holes arrive fromthe right. Once an electron-hole pair is formed, the spinsof the pair-partners undergo precession in the fields Ω and Ω , respectively, waiting to either recombine or tobypass each other and proceed along their respective cur-rent paths. For simplicity we choose the current paths inthe form of 1 D chains. This choice makes the adoptedmodel of transport very close to the “two-site” modelproposed in Ref. 15. The on-site dynamics of a pair withrecombination was studied in detail in previous sections.To utilize the results of Sect. III for the calculation ofcurrent, I , one has to incorporate the stages of formationand dissociation of pairs into the description of transport.In Fig. 4 the formation and dissociation are illustratedwith white double-sided arrows. The formation time forall four variants of initial states is assumed to be thesame, τ D . For simplicity we choose the average time forbypassing to be also τ D . Note that this choice does notlimit the generality of the description, provided that τ D islonger than the recombination time. The middle and thebottom portions in Fig. 4 illustrate the spin precession(blue arrows) and recombination (brown arrow) stages,which we studied earlier. Implicit in Fig. 4, is that thepair disappears either due to dissociation or by recom-bination before the next charge carrier arrives. Anotherway to express this fact is to state that the passage ofcurrent proceeds in cycles .Naturally, subsequent cycles are statistically indepen-dent. This allows one to express the current along a 1 D path through the average duration of the cycle, t . In-deed, N ≫ T N = t + t + · · · + t n .For large N , this net time acquires a gaussian distribu-tion centered at T N = N t . Correspondingly, the current,
N/T N , saturates at the value I = 1 t . (43)Note, that Eq. (43) constitutes an alternative approachto solving the system of rate equations for two-site model,as in Ref. 15, or to solving numerically the steady-statedensity-matrix equations, as in Ref. 16. Note also, thatEq. (43) is applicable to such singular realizations as softpairs, while previous approaches are not. For detaileddiscussion of this delicate point see Ref. 25.The remaining task is to express t via the average re-combination time, h t R i and τ D . For a typical pair in theregime of slow recombination h t R i is given by Eq. (38)upon substitution of Eq. (21). Using this expression weget for average duration of the cycle t = τ D + 14 × τ (cid:16) − Ω · Ω | Ω || Ω | (cid:17) + τ D + 2 × τ (cid:16) Ω · Ω | Ω || Ω | (cid:17) + τ D . (44)The first term captures the formation of the pair, while1 /τ D in the denominators describes the bypassing. In-deed, if recombination times are ∼ τ , one can neglect1 /τ D in the denominators. On the other hand, as the brackets in denominators in Eq. (44) turn to zero, whichcorresponds to anomalously slow recombination, the sec-ond term becomes τ D . Similarly, for slow recombinationwith soft pairs, using Eq. (39) we get t = τ D + 14 × τ (cid:16) − Ω · Ω | Ω || Ω | (cid:17) + τ D + 1 τ (cid:16) | Ω + Ω | | Ω + Ω | +4 | Ω × Ω | (cid:17) + τ D + 1 τ (cid:16) ( | Ω | −| Ω | ) | Ω + Ω | (cid:17) + τ D . (45)Finally, in the regime of fast recombination one shoulduse Eq. (42) for h t R i . This leads to the following expres- sion for tt = τ D + 14 τ + τ D + 1 τ (cid:16) ( | Ω | −| Ω | ) | Ω + Ω | (cid:17) + τ D + 2 × τ (cid:16) | Ω × Ω | | Ω + Ω | (cid:17) + τ D . (46)Obviously, the dependence of current on external fieldis encoded in Eqs. (44)-(46) via the frequencies Ω = B + b e and Ω = B + b h . The observable is the currentaveraged over realizations of the hyperfine fields b e and b h . This averaging is performed in the next section. V. AVERAGING OVER HYPERFINE FIELDSA. Averaging in the slow-recombination regime
Our basic assumption is that the time, τ D , of forma-tion and dissociation of a pair is much bigger than therecombination time, τ . Only under this condition thepair will exercise the spin dynamics. Using the relation τ D ≫ τ , we can simplify the expression Eq. (44) for t ofa typical pair t = τ D + τ − (cid:16) Ω · Ω | Ω || Ω | (cid:17) . (47)We can also rewrite the current in the form I = τ D − δI t ( B ), where the field-dependent correction is definedas δI t ( B ) = ττ D − (cid:16) Ω · Ω | Ω || Ω | (cid:17) + ττ D (48)As we will see below, the significant change of δI t with B takes place in the domain where B is much bigger thanthe hyperfine field. Therefore, we expand Eq. (48) withrespect to | b e | /B and | b h | /B . The principal ingredient ofthis step is the expansion of denominator | Ω | | Ω | − ( Ω · Ω ) ≈ B " | b e − b h | − (cid:18) b e · B B − b h · B B (cid:19) . (49)Assuming identical Gaussian distributions of b e , b h P ( b i ) = 1( πb ) / exp( −| b i | /b ) , (50)and choosing the z -direction along B we get h δI t ( B ) i = B ττ D * b x − b x ) + ( b y − b y ) + ττ D B + . (51) . . . . . . . . . − − − − F x x . x F ( x ) FIG. 5: (Color online). Blue line: Magnetic field response, δI t ( B ), for the “parallel-antiparallel” blocking mechanism, isplotted from Eq. (54) in the units 1 /τ D versus dimension-less magnetic field B/B c . Green line: fit with conventionallineshape of OMAR, x / (0 . x ). The next step is averaging Eq. (51) over the remainingfour components of the hyperfine fields. It is easiest toperform this integration by switching to b ± b and in-troducing the polar coordinates. The integrations overthe sum and over the polar angle are elementary. Theresult can be cast in the form h δI t ( B ) i = 1 τ D F (cid:18) BB c (cid:19) , (52)where the characteristic field B c is given by B c = (cid:18) τ D τ (cid:19) / b . (53)The form of the function F is the following F ( x ) = 2 x ∞ Z du uu + x e − u = x e x E ( x ) , (54)where E ( z ) is the exponential integral function. FromEq. (53) we see that relation τ D ≫ τ ensures that B c ≫ b , so that the expansion Eq. (49) of δI t ( B ) with respectto hyperfine fields is justified.The magnetoresistance Eq. (52) is plotted in Fig.5. We note that the shape, being a single-parameterfunction, F ( x ), can be very closely approximated with x / (0 . x ). This approximation, which is also plottedin Fig. 5, represents a standard fitting function for ex-perimentally measured magnetoresistance. It can be seenthat at x ≪ F ( x ) from theapproximation. This is due to singular behavior of F ( x )at small arguments. This singularity translates into thefollowing behavior of δI t ( B ) δI t ( B ) ≈ b
Soft pairs are responsible for the second and thirdterms in the brackets of Eq. (45) for t . The second termbecomes big when the sum, Ω + Ω , becomes anoma-lously small. Still it cannot dominate over the contribu-tion from the first term for the following reason. When Ω + Ω is small, the expression in the parenthesis of thesecond term behaves as ( Ω + Ω ) / | Ω | . At the sametime, for small Ω − Ω , the expression in the paren-thesis of the first term behaves as ( Ω − Ω ) / | Ω | . Instrong fields, the second expression is smaller than thefirst, leading to the larger δI , while in weak fields thetwo expressions give the same contribution to δI .The third term in Eq. (45) captures the contribu-tion of the slow modes to the current. Below we willstudy whether the averaging of this term over hyperfinefields can dominate over the “bipolaron” magnetic-fieldresponse given by Eq. (55).Prior to performing averaging, we rewrite the currentas I = τ D − δI s ( B ), like we did above. In the soft-pairs-dominated regime the expression for δI s ( B ) takesthe form δI s ( B ) = 1 τ D
11 + ( | Ω | −| Ω | ) | Ω + Ω | τ τ D . (56)For a typical configuration with | Ω | ∼ | Ω | , the secondterm in denominator can be estimates as | Ω | τ τ D , sothat it is large in the slow-recombination regime. This iswhy the soft pairs with( | Ω | − | Ω | ) ∼ √ τ τ D (57)give the major contribution to the average δI s ( B ). Thelatter fact allows one to simplify the averaging proce-dure. Namely, one can use the fact that for ǫ ≪ ǫǫ + x can be replaced by πδ ( x ). Thus, theexpression to be averaged can be rewritten in the form δI s ( B ) = πτ D √ τ τ D δ (cid:18) | Ω | − | Ω | | Ω + Ω | (cid:19) . (58)The form Eq. (58) suggests that characteristic magneticfield determined from zero of the δ -function is B ∼ b ,and yields the estimate 1 /τ / τ / D b for δI s ( B ). To com-pare the contribution of soft pairs to that of typical pairs0 . . . . . . . . . . . h δ I ( η , B ) i − h δ I ( η , ) ih δ I ( , ) i Bb η = . η = . η = . η = . η = . η = . η = . η = . η = . η = . η → √ x x + . η → FIG. 6: (Color online). Magnetic field response for the “soft-pair” mechanism is plotted from Eq. (66) versus magneticfield in the units of the hyperfine field b for different valuesof the asymmetry parameter η . Inset: fit of the response inthe limit of strong asymmetry with conventional lineshape ofOMAR, √ x / (0 .
23 + x ). this estimate should be compared to Eq. (55) taken at B ∼ b . Soft pairs dominate if the condition r τ D τ ≫ b τ (59)is met. Since τ D is much bigger than τ , this conditionis compatible with the condition, b τ ≫ δ -function, as we didfor soft pairs, is not permissible. This follows, e.g. , fromEq. (53) which suggests that the characteristic field B c is much bigger than b . Replacement of the denominatorin Eq. (48) by a δ -function would automatically fix thecharacteristic field at B ∼ b .In averaging of Eq. (58) over hyperfine configurations,we will assume from the outset that the characteristichyperfine fields, b and b , for the electron and hole aredifferent, so that h δI s ( B ) i = 1 π b b p τ τ D Z d b e Z d b h δ (cid:18) | b e + B | − | b h + B | | b e + b h + 2 B | (cid:19) exp (cid:18) − | b e | b − | b h | b (cid:19) . (60)Subsequent analysis will indicate that different b and b is a necessary condition for δI s to exhibit B -dependence. The six-fold integral Eq. (60) can be reduced to asingle integral in three steps. As a first step, we introducenew variables v = b e − b h and u = b e + b h + 2 B , so thatEq. (60) acquires the form h δI s ( B ) i = 18 π b b p τ τ D Z d u Z d v | u | δ ( u · v ) × exp (cid:0) − α ( u − B ) + β ( u − B ) · v − α | v | (cid:1) , (61)with parameters α and β defined as α = 14 (cid:18) b + 1 b (cid:19) , β = 12 (cid:18) b − b (cid:19) . (62)As a second step, we perform integration over the vector v . The reason why this integration can be carried outanalytically is that, upon choosing the z -direction along u , the δ -function fixes v z to be zero. The remaining twointegrals over v x and v y are simply gaussian integrals, sowe get h δI s ( B ) i = 18 πb b α p τ τ D Z d u exp (cid:20) − α ( u − B ) + β α (cid:18) | B | − ( B · u ) | u | (cid:19)(cid:21) . (63)To perform the integration over u , we switch to sphericalcoordinates with polar axis along B . Then the integra-tion over azimuthal angle reduces to multiplication by2 π . The third step is the integration over the polar anglein Eq. (63). We have h δI s ( B ) i = e − α (cid:16) − β α (cid:17) B b b α p τ τ D ∞ Z du u e − αu π Z dθ sin θ exp (cid:18) αuB cos θ − β α B cos θ (cid:19) . (64)Now we note that the integral over θ can be expressedvia the error-functions in the following way Z − dx e − A x + Cx = √ π A e C A (cid:20) erf (cid:18) A + C A (cid:19) + erf (cid:18) A − C A (cid:19)(cid:21) . (65)We are left with a single integral over u , which can becast in the form h δI s ( B ) i = √ πe − α (cid:16) − β α (cid:17) B b b p τ τ D √ αβB ∞ Z du u exp (cid:18) − α (cid:18) − α β (cid:19) u (cid:19) (cid:20) erf (cid:18) βB √ α + 2 α / uβ (cid:19) + erf (cid:18) βB √ α − α / uβ (cid:19)(cid:21) . (66)1 . . . . . . . . .
910 1 2 3 4 5 6 7 8 9 10 h δ I ( B ; κ ) i / h δ I ( κ = ) i B/b . . κ = . κ = . κ = . κ = . κ = . κ = . κ = . κ = . κ = . κ = . κ = . . e − . x κ = . FIG. 7: (Color online) Magnetic field response caused by thedifference, in the g -factors of electron and hole is plotted fromEq. (73) for several values of relative difference, κ . Upper in-set illustrates that the shape of the response is near-gaussian.Lower inset illustrates that at κ close to 1 the shape of theresponse develops a maximum. C. Analysis of Eq. (66)
At this point we make an observation that for b = b ,which is equivalent to β = 0, magnetic field drops out ofEq. (66). The easiest way to see it is to set β = 0 at theearlier stage of calculation, namely in Eq. (63) h δI s ( b = b ) i = 18 πb b α p τ τ D Z d u e − α ( u − B ) , (67)which is clearly independent of B after a simple coordi-nate shift. If we set b = b , then δI s is given by h δI s ( b = b ) i = r π τ τ D b , (68)in agreement with the qualitative estimate above.Magnetic field dependence of h δI s i emerges already atsmall values of asymmetry parameter defined as η = 1 − b b . (69)This is illustrated in Fig. 6, where h δI s ( η, B ) i−h δI s ( η, i in the units of h δI s ( η = 0) i , given by Eq. (68), is plot-ted for several values of η . We see that, as η increases,the shape of the curves does not change much. For thesaturation value the analysis of Eq. (66) yields h δI s ( η, ∞ ) i − h δI s ( η, ih δI s ( η = 0) i = √ η (2 − η ) / . (70) The result Eq. (66) can be recast in the more con-cise form in terms of the Dawson function D ( x ) = e − x x R dt e t . The corresponding expression reads h δI s ( η, B ) ih δI s (0 , i = √ √ − η − (cid:18) η − η (cid:19) √ z D (cid:18) z √ − η (cid:19) , (71)where we have introduced z = B/b .In the limit of strong asymmetry, when η is close to 1,one gets a simple analytical expression for h δI s ( B ) ih δI s ( η = 1 , B ) ih δI s ( η = 0) i =2 (cid:18) Bb (cid:19) Z − dx √ x exp " − (cid:18) Bb (cid:19) (1 − x ) . (72)At small B the ratio Eq. (72) behaves quadratically,while at large B it saturates as √ (cid:16) − b B (cid:17) . Over-all, similarly to I t ( B ), magnetoresistance Eq. (72) canbe closely approximated with √ x / (0 .
23 + x ), as illus-trated in Fig. 6. D. Inequivalence of electron and hole g -factors In the previous subsection we demonstrated that ex-ternal magnetic field drops out from the general expres-sion Eq. (60) when the variances b and b are equal.Here we note that averaging does not eliminate the B -dependence even when b = b , as long as the g -factorsof the pair partners are different. Incorporating g and g into Eq. (60) is straightforward and amounts to mul-tiplying b e + B by 1 + κ , while b h + B is multiplied by1 − κ , where κ is the relative difference in the g -factors.The three steps leading from Eq. (60) to Eq. (66) areexactly the same as for κ = 0. Finite κ modifies boththe prefactor in the integral Eq. (66) and the argumentsof the error functions in the integrand. It is convenientto analyze the magnetic field response by considering theratio h δI s ( B ; κ ) i / h δI s ( κ = 0) i , where the denominator isgiven by Eq. (68). h δI s ( B ; κ ) ih δI s ( κ = 0) i = exp (cid:16) − z κ (cid:17) √ κ ) κz ∞ Z du u e − ζu (cid:20) erf (cid:18) κ − κ ( z + γu ) (cid:19) + erf (cid:18) κ − κ ( z − γu ) (cid:19)(cid:21) , (73)2where z = B/b is the scaled magnetic field. For nota-tional convenience we introduced the κ -dependent terms ζ and γ , which are defined as ζ = 12(1 − κ ) (cid:18) − κ ) κ (1 + κ ) (cid:19) , (74) γ = 11 − κ (cid:18) − κ ) κ (1 + κ ) (cid:19) . (75)It is seen that the arguments of the error-functions aswell as the power in the exponent diverge in the limit κ →
1, i.e. when the g -factor of one pair-partner is zero.This divergence signifies that magnetic field response isweak for small (1 − κ ). The underlying reason for this isthat the portion of soft pairs goes to zero if the levels ofone of the partners are not split by a magnetic field. InFig. 7 we plot the magnetic field response for differentvalues of κ . There are two noteworthy features of thisresponse. Firstly, the sign of response is opposite to thatfor inequivalent distributions of electrons and holes, seeFig. 6. Secondly, the shape of δI s ( B ) is not Lorentziananymore. In fact, this shape is close to Gaussian, as il-lustrated in the inset. Another peculiar feature of δI s ( B )which can be seen from Fig. 7 is that, for κ close to 1,the response δI ( B ) develops a bump. E. Averaging in the fast-recombination regime
Turning to Eq. (46) for t in the fast-recombinationregime we notice that the second term in the squarebrackets has exactly the same form as the contribution ofthe soft pairs to t in the slow-recombination regime, seeEq. (45). The underlying reason is that, similarly to softpairs, this second term also comes from the slow eigen-mode. The origin of this slow eigenmode, i.e. orthogonal-ization of S -mode to all the other states, was discussed indetail in Sect. IIc. Since the configurational averaging forsoft pairs was already carried out, we conclude that themagnetic field response in the fast-recombination regimeis simply described by Eq. (66).At this point we note that configurational averagingover slow pairs was based on the applicability of the con-dition b τ τ D ≫
1. Therefore, it is important that thiscondition is compatible with fast-recombination, b τ ≪
1, by virtue of a small parameter τ /τ D .In addition to the soft-pair contribution, Eq. (46)also contains a term with | Ω × Ω | in the denomi-nator. This term becomes large when Ω and Ω arecollinear. However, the statistical weight of these con-figurations is smaller than the statistical weight of thesoft-pair contribution. Indeed, in order for the term with | Ω × Ω | in denominator to become large, the anglebetween the vectors Ω and Ω should be restricted to θ ∼ /b √ τ τ D ≪
1. In course of configurational averag-ing, the integral, R dθ sin θ . . . , emerges which is small as θ . b τ b τ D b ττ D = 1 q τ D τ = b τI II FIG. 8: (Color online). Different domains on the plane( b τ D , b τ ) illustrate the regions where different OMARmechanisms dominate. The is no OMAR in the white do-mains. The pink domain corresponds to slow recombination,and OMAR is given by Eq. (52). In both the upper and thelower parts of the gray domain the OMAR is dominated bysoft pairs and is described by Eq. (66). The green line dividesthe gray domain into subregions where the recombination isslow (upper part) and fast (lower part). The boundaries ofthe domains are: b τ = b τ D , and b τ = ( b τ D ) / . We now turn to the limit of very weak hyperfine fieldsfor which the parameter b τ τ D is small. One may expectthat magnetic field response is suppressed in this domain.What we demonstrate below is that this suppression isanomalously strong. Namely, the first term of the expan-sion of Eq. (46) with respect to b τ τ D does not containthe external field at all . This first term has the form t − τ D = − τ τ D (cid:20) ( | Ω | − | Ω | ) | Ω + Ω | + 4 | Ω × Ω | | Ω + Ω | (cid:21) . (76)To realize that B drops out of the expression in the squarebrackets it is convenient to first replace | Ω × Ω | by | Ω | | Ω | − ( Ω · Ω ) and then use the identity | Ω + Ω | | Ω − Ω | = ( | Ω | + | Ω | ) − Ω · Ω ) . (77)This leads to a drastic simplification of Eq. (76), whichassumes the form t − τ D = − τ τ D | Ω − Ω | . (78)Since | Ω − Ω | = | b e − b h | , the magnetic field drops outof t in the first order in τ τ D b . VI. CONCLUDING REMARKS (i) Our findings can be summarized in the form of do-mains on the plane ( b τ D , b τ ), as shown in Fig. 8.The fact that for small b τ τ D the OMAR responseis absent is reflected in Fig. 8 by leaving the domain3lying below the hyperbola uncolored. Large hyper-fine fields, b τ >
1, correspond to slow recombina-tion. As we have demonstrated above, the OMARfor b τ > I and II , are colored in Fig. 8 by pink and gray, re-spectively. The domains are separated by the curve b τ = ( b τ D ) / . Eq. (52) describes OMAR in thedomain I , while in the domain II Eq. (66) applies.Note that in the domain II only the part above thegreen line corresponds to slow recombination. Thepart below the green line corresponds to fast re-combination, but Eq. (66) applies in both domains.The diagram describes the regimes of OMAR in lowapplied fields, B ∼ b . As B increases above b , thegray domain shrinks.(ii) The OMAR response from the soft pairs relies ex-clusively on the asymmetry between electron andhole. The evidence in favor of such an asymme-try was inferred in Ref. 27 from the analysis ofmagnetic-resonance data in organic devices. In Ref.27, the ratio b /b was estimated to be close to 3,which leads to the value of the asymmetry param-eter η ≈ .
9. Note, that bipolaron mechanism isinsensitive to the asymmetry between electron andhole.(iii) “Parallel-antiparallel” mechanism of Ref. 15 yieldsthe OMAR response on the level of rate equationswith the transition rates calculated from the goldenrule . The applicability of this treatment requiresthat the separation of Zeeman levels is large com-pared to their widths. On the other hand, theOMAR response based on soft pairs, studied in thepresent paper, comes entirely from pairs for whichthe Zeeman levels are almost aligned. This requiresone to go beyond the golden rule. Previously, a sim-ilar situation was encountered by M. Schultz andF. von Oppen in the study of transport through ananostructure with almost degenerate levels. Therole of spin-selective recombination was played bycoupling to the leads which was strongly differentfor symmetric and antisymmetric combinations ofthe wave functions. M. Schultz and F. von Oppenpointed out that when two levels are closer in en-ergy than the width of each of them, then the con-ventional rate-equation-based description is insuffi-cient.On the physical level, the near-degeneracy impliesthat some spin configuration is preserved duringmany precession periods, i.e. the dynamics is im-portant. To account for dynamics, it is intu-itively appealing to take the result of Schulten andWolynes, Eqs. (2)-(3) , and multiply it by a fac-tor describing exponential decay of population ofstates due to recombination. Such an approach wasadopted in Ref. 22. What this approach misses is the feedback of recombination on the pair dynam-ics. It is the central message of the present paperthat this effect is strong in certain regimes, sincefeedback creates long-living modes.(iv) The “parallel-antiparallel” mechanism of Ref. 15 isbased on the picture of incoherent hopping of oneof the charge carriers on the site already occupiedby the other carrier. We considered the transportmodel applicable for bipolar system where the pas-sage of current is due to recombination of electronsand holes. However, the principal ingredients ofboth models are the same: (a) in both transportmodels the spins of the carriers precess in their effec-tive magnetic fields, the precession being governedby the same Hamiltonian Eq. (1); (b) the passageof current is the sequence of cycles, only one stepof each cycle is sensitive to the spin precession; (c)whether it is a hop or recombination, it occurs onlyfrom the S -spin configuration; (d) if either the hopor recombination act takes too long, the carriers by-pass each other.(v) Both the “parallel-antiparallel” pairs and soft pairscreate the OMAR response by blocking the current.The origin of this blocking is completely differentfor the two mechanisms. In the former, the cur-rent is blocked due to collinearity of full fields forthe pair-partners, while for the latter the block-ing is due to coincidence of their absolute values.In general, both contributions are present in thefast-recombination regime. The contribution of softpairs in this regime dominates by virtue of theirstatistical weight.(vi) Another distinctive feature of the soft-pairs mech-anism follows from Eq. (56). It contains a combi-nation ( | Ω | − | Ω | ) in the denominator. As theprecession frequencies change with external field, B ,the pair undergoes evolution from typical to soft(when | Ω | = | Ω | ) and back to typical. Impor-tantly, this evolution takes place within a narrowinterval of B , so that at a given B only certainsparse pairs contribute to the current. As demon-strated in Ref. 28, this redistribution of soft pairsgives rise to mesoscopic features in I ( B ) in smallsamples.(vii) We have demonstrated above that regardless ofwhether the OMAR is due to blocking caused by“parallel-antiparallel” configurations, as in Ref. 15,or due to soft pairs, the shape of the response isalways close to B / ( B + B c ). This result wasobtained under the assumption that τ and τ D are fixed . If the values of τ and τ D are broadly dis-tributed, then the adequate description of trans-port should be based on the percolative approach .However, within our minimal model, the current isthe sum of partial currents through the chains, seeFig. 3. Then, with wide spread in cycle durations,4 t , the current will be limited by pairs with longest t present in each chain. Acknowledgments
We are grateful to Z. V. Vardeny and E. Ehrenfreundfor illuminating discussions. This work was supported byNSF through MRSEC DMR-1121252 and DMR-1104495.
Appendix A: Time Evolution and the SchrodingerEquation
In this Appendix we sketch a formal derivation of Eqs.(19) and (37) starting from the Liouville equation for thedensity operator, b σ , ∂ b σ∂t = − i [ b H , b σ ] + b L ( b σ ) , (A1)where the term b L ( b σ ) describes relaxation, which in ourcase is recombination from S to the ground state, G .The ground state with energy −E is included into thebare Hamiltonian b H = b H + b H G (A2)= (cid:16) b S · B + b S · B (cid:17) − E | G i h G | . (A3)Then the operator b L ( b σ ) cast into conventional Lindbladform reads b L ( b σ ) = 12 Γ (2 | G i h S | b σ | S i h G | − b σ | S i h S | − | S i h S | b σ ) , (A4)where Γ = τ − is the inverse recombination time.Denote with i , k different spin configurations of thepair prior to recombination. The form Eq. (A4) of thedissipation ensures independence of the elements of thedensity matrix with subindices i , k from the elementscontaining subindex G . This decoupling follows from thefull system of the equations of motion ∂σ GG ∂t = Γ σ SS , (A5) ∂σ Gk ∂t = −E σ Gk − X i σ Gi H ik −
12 Γ σ Gk δ Sk , (A6) ∂σ ik ∂t = − i [ b H , b σ ] ik −
12 Γ { b σ, | S i h S |} ik . (A7)Eq. (A7) couples only the elements of 4 × ρ , so that Eq. (A7) represents equationof motion for ρ . These equations can be rewritten in theform similar to Eq. (A1) ∂ b ρ∂t = − i [ b H, b ρ ] + b L ( b ρ ) , (A8) with dissipation term redefined as b L ( b ρ ) = − Γ { b ρ, | S i h S |} . To derive Eq. (19), we searchfor solution of Eq. (A8) in the form ρ ( t ) = | ψ ( t ) i h ψ ( t ) | , (A9)and find that ψ ( t ) must satisfy the following non-hermitian Schr¨odinger equation i ∂∂t | ψ ( t ) i = b H ′ | ψ ( t ) i , (A10)where b H ′ is defined as b H ′ = b H − i Γ2 | S i h S | . The factthat decoupling Eq. (A9) is valid follows from a straight-forward calculation i ∂∂t | ψ i h ψ | = (cid:18) i ∂∂t | ψ i (cid:19) h ψ | + | ψ i (cid:18) i ∂∂t h ψ | (cid:19) (A11)= H ′ | ψ i h ψ | − | ψ i h ψ | ( H ′ ) † (A12)= (cid:18) H − i Γ2 | S i h S | (cid:19) | ψ i h ψ |− | ψ i h ψ | (cid:18) H + i Γ2 | S i h S | (cid:19) (A13)= [ H, | ψ i h ψ | ] − i Γ2 {| S i h S | , | ψ i h ψ |} . (A14)Now Eq. (19) immediately emerges as an equation foreigenvalues of the operator b H ′ . Appendix B: Derivation of Eq. (37)
To derive Eq. (37) for recombination time from ran-dom initial state, we first find the expression for recombi-nation time, t ψ , from a given initial state, ψ , in terms ofthe solution of Eq. (A8) for ρ ( t ) complemented with con-dition ρ (0) = | ψ i h ψ | . The expression for t ψ in termsof the full density matrix b σ ( t ) reads t ψ = Z ∞ (cid:18) dt ∂σ GG ∂t (cid:19) t. (B1)The meaning of the expression in the brackets is the prob-ability that recombination took place between t and t + dt .The expression for t ψ in terms of ρ ( t ) follows from therelation σ GG + Tr ρ = 1 . (B2)Performing integration by parts, we obtain t ψ = Z ∞ dt Tr ρ ( t ) . (B3)To find the recombination time h t R i from the randominitial state the time t ψ should be averaged over initialstates. One way to perform this averaging is to fix acertain orthonormal basis, Φ k , expand ψ as ψ = X k c k Φ k , (B4)5and express t ψ as bilinear form in c k . This yields t ψ = Z ∞ dt Tr h b U ( t ) ψ ∗ ψ b U † ( t ) i (B5)= Z ∞ dt Tr " b U ( t ) "X k c k Φ k k ′ c k ′ ∗ Φ k ′ ∗ U † ( t ) (B6)= X k,k ′ c k c k ′ ∗ Z ∞ dt Tr h b U ( t )Φ k Φ k ′ ∗ b U † ( t ) i , (B7)where b U ( t ) is the non-unitary evolution operator. Nowthe averaging over initial conditions reduces to averag-ing over c k according to the rule h c k c k ′ ∗ i = δ k,k ′ . Thisaveraging is straightforward leading to h t R i = 14 X k Z ∞ dt Tr h b U ( t )Φ k Φ k ∗ b U † ( t ) i (B8)The remaining task is to express the sum, Eq. (B8), interms of eigenvalues and eigenvectors of a non-hermitianSchr¨odinger equation, Eq. (A10). To accomplish thistask we will use the expansion of the solutions ψ k of Eq.(A10), which we, for brevity, denote with | λ k i , in termsof the orthonormal basis Φ k , which we denote with | k i .In terms of these new notations Eq. (A10) and thetime evolution operator can be written as b H ′ | λ j i = λ j | λ j i , b U ( t ) | λ j i = e − iλ j t | λ j i . (B9)It is also convenient to introduce a matrix, b d , whichrelates the elements of the basis to the solutions of Eq.(A10). Namely, | k i = X l d kl | λ l i . (B10)Substituting Eq. (B10) into Eq. (B9), we find b U ( t ) | k i = X l d kl b U ( t ) | λ l i = X l d kl e − iλ l t | λ l i . (B11)Next we introduce, b g , which is the matrix of scalar prod-ucts g ij = h λ i | λ j i . (B12)Using the definitions Eq. (B10) and Eq. (B12) we express h t R i , defined by Eq. (B8), in terms of the matrices b d and b g h t R i = 14 X k ∞ Z dt Tr h b U ( t ) | k i h k | b U † ( t ) i , (B13)= 14 X k Z ∞ dt Tr " b U X l d kl | λ l i ! × X m d ∗ km h λ m | ! b U † , (B14)= 14 X klm ∞ Z dt e − i ( λ l − λ ∗ m ) t d kl d ∗ km Tr [ | λ l i h λ m | ] , (B15)= 14 X lm i ( λ l − λ ∗ m ) g ml X k d kl d ∗ km ! . (B16) In the last identity we have isolated the combination ofthe elements of the matrix b d . The reason is that thiscombination can be cast in the form X k d kl d ∗ km = g − ∗ ml . (B17)To prove the latter identity, we start from the matrixrelation h λ l | i i = X j d ij h λ l | λ j i = X j d ij g lj , (B18)and invert it to obtain d ij = X l g − jl h λ l | i i . (B19)Next we complex conjugate both sides of Eq. (B19)which yields d ∗ ij = X l g − ∗ jl h λ l | i i ∗ = X l g − ∗ jl h i | λ l i . (B20)Now, the identity Eq. (B17) emerges as a result ofstraight-forward calculation X k d kl d ∗ km = X k X j g − lj h λ j | k i X n g − ∗ mn h k | λ n i ! , (B21)= X jk g − lj g − ∗ mn h λ j | X k | k i h k | ! | λ n i (B22)= X jn g − lj g − ∗ mn g jn (B23)= g − ∗ ml . (B24)Finally, substituting Eq. (B17) into Eq. (B16), we arriveat Eq. (37) of the main text.6 E. L. Frankevich, I. A. Sokolik, D. I. Kadyrov, and V. M.Kobryanskii, Pis’ma Zh. Eksp. Teor. Fiz. , 401 (1982)[JETP Lett. , 488 (1982)]. E. L. Frankevich, A. A. Lymarev, I. Sokolik, F. E. Karasz,S. Blumstengel, R. H. Baughman, and H. H. H¨orhold,Phys. Rev. B , 9320 (1992). K. M. Salikhov, Y. N. Molin, R. Z. Sagdeev, and A. L.Buchachenko, in
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