Nonequilibrium Green's function theory for predicting device-to-device variability
GGreen’s function theory for predicting device-to-device variability
Yu Zhu, ∗ Lei Liu, and Hong Guo
2, 1 NanoAcademic Technologies Inc., 7005 Blvd. Taschereau, Brossard, QC, J4Z 1A7 Canada Department of Physics, McGill University, Montreal, QC, H3A 2T8, Canada (Dated: October 29, 2018)Due to random dopant fluctuations, the device-to-device variability is a serious challenge to emerging na-noelectronics. In this work we present theoretical formalisms and numerical simulations of quantum transportvariability, based on the Green’s function technique and the multiple scattering theory. We have developed ageneral formalism using the diagrammatic technique within the coherent potential approximation (CPA) thatcan be applied to a wide range of disorder concentrations. In addition, we have developed a method by usinga perturbative expansion within the low concentration approximation (LCA) that is extremely useful for typicalnanoelectronic devices having low dopant concentration. Applying both formalisms, transport fluctuations dueto random impurities can be predicted without lengthy brute force computation of ensemble of devices struc-tures. Numerical implementations of the formalisms are demonstrated using both tight-binding models and firstprinciples models.
PACS numbers: 73.63.-b, 73.23.-b, 72.80.Ng, 31.15.A-
I. INTRODUCTION
A very important yet difficult issue of electronic devicephysics is to be able to predict fluctuations in quantum trans-port properties due to atomic disorder . In existing andemerging field-effect transistors with a channel length of ∼ nm or so, a serious source of property unpredictability isthe random dopant fluctuation (RDF). RDF comes from theparticular microscopic arrangement of the small number ofdopant atoms inside the device channel. Experimentally itis extremely difficult – if not absolutely impossible, to con-trol the precise location of each dopant atom, therefore trans-port properties vary from one device to another. It was evenpointed out that nanowire transistors can suffer from RDF inthe source/drain extension region even if the channel is dopantfree . The device-to-device variability is in fact a generalphenomenon for device structures in the nano-meter scalewhich compromises device performance and circuit function-ality. From the theoretical point of view, incorporating disor-der and randomness in nano-electronics modeling is of greatimportance . In particular, one is interested in predicting notonly the average value of the transport property (e.g. conduc-tance) but also the variance of it.The device-to-device variability has so far been investigatedby statistical analysis of large number of simulations. For in-stance Reil et al carried out classical drift-diffusion simula-tions for an ensemble of dopant configurations under thecombined influence of RDF and line edge roughness . Mar-tinez et al did effective-mass nonequilibrium Green’s function(NEGF) simulations of an ensemble of dopant configura-tions to analyze statistical variability of quantum transport ingate-all-around silicon nanowires . The contrast of the size ofthe statistical ensemble clearly shows the difficulty of quan-tum simulations. The difficulty in brute force computationbecomes much more severe in full self-consistent atomisticmodeling (as opposed to effective-mass modeling) such as theNEGF based density functional theory (DFT) . There is anurgent need to develop viable theoretical methods that doesnot rely on brute force computation for predicting the device- to-device variability. It is the purpose of this work to presentsuch a formalism.We shall report a new theoretical approach to directly cal-culate statistical variations of quantum transport due to RDFwithout individually computing each and every impurity con-figuration by brute force. Our theory is composed of two for-malisms: one is general but more complicated and the other isspecialized but much simpler. The two formalisms are basedon the Green’s function technique and the multiple scatter-ing theory. The first formalism builds on coherent potentialapproximation (CPA) and can be applied to a wide range ofimpurity concentrations. The second formalism builds on thelow concentration approximation (LCA) and is extremely use-ful for situations involving low impurity concentration whichis often the case for realistic semiconductor devices. Our the-ory and implementation have been checked by both analyticaland numerical verification.The basic physical model of a two-probe quantum co-herent nanoelectronic device is schematically shown in anyone of the sub-figures of Fig.1, which consists of a centralchannel region sandwiched by the left and right semi-infiniteelectrodes . The electrodes extend to reservoirs at z = ±∞ where bias voltages are applied and electric current measured.We assume that the RDF occurs in the channel region of thesystem and each sub-figure in Fig.1 represents one dopantconfiguration. Clearly, due to different locations of the dopantatoms, every device exhibits slightly different transport behav-ior leading to the device-to-device variability. The transportcurrent flowing through the device can be expressed in termsof the transmission coefficient T (hereafter atomic units areassumed, e = (cid:126) = 1 ), I = (cid:90) dE π T ( E ) [ f L ( E ) − f R ( E )] , (1)where E is the electron energy, f L ( E ) and f R ( E ) are theFermi functions of the left/right electrodes. Without RDF, theelectric current I is a definite number for a given bias voltage.In the presence of RDF, I depends on the particular impurityconfiguration thus varies from one configuration to another. a r X i v : . [ c ond - m a t . m e s - h a ll ] N ov FIG. 1: Ensemble of two-probe devices with various disorder configurations. In each sub-figure, the left and right electrodes extend to z = ±∞ , respectively. The black dots are pure sites in the electrodes, the empty circles are pure sites in the central channel region, and thecrossed empty circles are disorder sites (dopant or impurity) in the channel region. By calculating a large ensemble of configurations one can de-termine an average current and its associated variance δI . Forour device model where RDF occurs inside the channel re-gion, δI is obtained in term of transmission fluctuation, δT ,as follows δI ≈ (cid:90) dE π δT ( E ) [ f L ( E ) − f R ( E )] . (2)By definition, the transmission fluctuation δT is obtained fromthe RDF ensemble average, δT ≡ (cid:113) T − T , (3)where · · · refers to averaging over the disorder configura-tions. Notice that the transmission coefficient T can be ex-pressed in terms of Green’s functions. As a result the calcu-lation of T involves evaluating a 2-Green’s function correla-tor G · G . The calculation of transmission fluctuation δT ( E ) which needs the quantity T involve a 4-Green’s function cor-relator G · G · G · G .In the literature, a well known technique called coherentpotential approximation (CPA) is available to evaluate dis-order average of a single Green’s function G . The CPA tech-nique was generalized to evaluate 2-Green’s function cor-relators G · G and 3-Green’s function correlators G · G · G (albeit in other contexts). More recently, the generalized CPAtechnique for calculating the 2-Green’s function correlatorhas been applied to study transmission and nonequilibriumquantum transport in disordered systems. This work will ad-dress how to evaluate 4-Green’s function correlator and applythe technique to study device variability.In a very recent manuscript , Zhuang and Wang carriedout an analysis of conductance fluctuation and shot noise in graphene by using a direct expansion approach. To some ex-tent, their approach is complementary to the methods pre-sented in this work with respect to accuracy and efficiency.Finally, there are large bodies of literature in mesoscopicphysics to analyze such issues as the universal conductancefluctuation in bulk systems using the Kubo formula and δ -likeshort range impurity potentials . In contrast, the goal of thiswork is to formulate a theoretical approach for calculating thetransmission fluctuation of two-probe systems where the dis-order scattering is due to impurity atoms as opposed to δ -likemodels.The rest of the paper is organized as follows. Section IIreviews the multiple scattering theory of the t-matrix formal-ism. Section III presents the first formalism, i.e. the CPAdiagrammatic technique for calculating transmission fluctua-tion. Section IV presents the second formalism, i.e. the LCAperturbative expansion technique for calculating transmissionfluctuation. Section V discusses a special but important situ-ation where the device structure is periodic in transverse di-mensions. Section VI presents some miscellaneous technicalissues of the theory. Section VII presents three examples asapplications of the CPA and LCA formalisms. Finally, the pa-per is concluded with a brief summary in Section VIII. Sometechnical details are enclosed in the two appendices. II. THE T-MATRIX FORMALISM
To simulate disorder sites in the central region, the on-site energies are assumed to be discrete random variables.Namely, on a disorder site- i the on-site energy ε i can takethe value ε iq with the probability x iq where q = 1 , , · · · in-dicating the possible atomic species on that site and the nor-malization requires (cid:80) q x iq = 1 .For a given disorder configuration { ε i } , the transmissioncoefficient T ( E ) can be derived in terms of the Green’s func-tions of the central region T ( E ) = Tr [ G r ( E )Γ L ( E ) G a ( E )Γ R ( E )] , (4)where G r,a are the retarded and advanced Green’s functions, Γ L,R are the linewidth functions of the left and right elec-trodes. The retarded Green’s function can be derived as G r ( E ) = [ E − H ( { ε i } ) − Σ r ( E )] − , (5)where H ( { ε i } ) is the Hamiltonian of the central region whoseimpurity configuration is { ε i } ; Σ r is the retarded self-energyto take into account the influences of the semi-infinite elec-trodes on the central region. The line-width function Γ inEq.(4) is related to the self-energy Γ β ( E ) = i (cid:2) Σ rβ ( E ) − Σ aβ ( E ) (cid:3) , (6)where β = L, R labels the left or right electrode and Σ r ( E ) =Σ rL ( E ) + Σ rR ( E ) is the total retarded self-energy. The ad-vanced Green’s function and self-energy are Hermitian conju-gates of their retarded counterparts, G a ( E ) = G r ( E ) † , Σ aβ ( E ) = Σ rβ ( E ) † . To determine δT by Eq.(3), one needs to calculate the con-figuration averaged quantities T and T . While vertex correc-tion technique have been applied successfully to calculate T ,the quantity T turns out to be extremely difficult to calculateand the goal of this work is to derive a necessary formulationfor it. Since our approach is based on the t-matrix formalism,in the rest of this section, we briefly review the well knownt-matrix formalism following Ref.12.Recall that H is the Hamiltonian of central region whichcontains some disorder sites (see Fig.1). Divide H into twoparts, H and V , where H is the definite part of the Hamilto-nian and V is the random disorder potential, H = H + V, (7) V = (cid:88) i ˆ V i , (8)in which ˆ V i is the random potential of disorder site- i . ˆ V i is anearly all-zero matrix except for its i -th diagonal element ˆ V i = diag [0 , · · · , , V i , , · · · , , where V i is a discrete random variable which can take thevalue V iq with the probability x iq . V iq is related to the on-site energy ε iq by V iq = ε iq − ε i , where ε i is a site-dependentarbitrary constant. Due to different choices of (cid:8) ε i (cid:9) , the par-tition of H into H and V is not unique. We shall exploit thisfreedom and adopt different partitions for CPA and LCA (seenext two sections). With the partition Eqs.(7,8), the retarded Green’s functionof Eq.(5) can be expressed in terms of the unperturbed Green’sfunction G r and the t-matrix T r G r = G r + G r T r G r , (9)where G r and T r are defined as G r ≡ [ E − H − Σ r ] − , (10) T r ≡ V (1 − G r V ) − . (11)The t-matrix T r can be further expanded in terms of scat-tering amplitude ˆ t ri , T r = (cid:88) i ˆ t ri + (cid:88) i (cid:88) j (cid:54) = i ˆ t rj G r ˆ t ri + (cid:88) i (cid:88) j (cid:54) = i (cid:88) k (cid:54) = j ˆ t rk G r ˆ t rj G r ˆ t ri + · · · , (12)where ˆ t ri represents multiple disorder scattering on the site- i ˆ t ri ≡ ˆ V i + ˆ V i G r ˆ V i + ˆ V i G r ˆ V i G r ˆ V i + · · · = ˆ V i (cid:16) − G r ˆ V i (cid:17) − . (13)Similar to ˆ V i , ˆ t ri is also a nearly all-zero matrix except for its i -th diagonal element ˆ t ri = diag [0 , · · · , , t ri , , · · · , , where t ri is a random variable which can take the value t riq with the probability x iq . t riq is obtained as t riq ≡ V iq (cid:0) − G r ,ii V iq (cid:1) − , (14)in which G r ,ii means to take the i -th diagonal element of G r .Inserting Eq.(12) into Eq.(9), G r can be expanded in a se-ries of scattering terms: G r = G r + (cid:88) i G r ˆ t ri G r + (cid:88) i (cid:88) j (cid:54) = i G r ˆ t rj G r ˆ t ri G r + (cid:88) i (cid:88) j (cid:54) = i (cid:88) k (cid:54) = j G r ˆ t rk G r ˆ t rj G r ˆ t ri G r + · · · . (15)In a diagrammatic language, Eq.(15) can be represented byFig.2 in which the thick line represents G r , the thin line rep-resents G r , and the dotted line with a crossed dot represents ˆ t ri (random variable). It is required that adjacent ˆ t ri lines musthave different site indices. Clearly, a similar expansion can becarried out for the advanced Green’s function G a .The t-matrix expansion in Eq.(15) is rigorous. By insert-ing Eq.(15) and its advanced counterpart into Eq.(4) and itssquare, after averaging over disorder configurations, T ( E ) and T ( E ) can be derived as a summation of products com-posed of x iq , G r and G a , Γ L and Γ R , ˆ t riq and ˆ t aiq . In principle,one can calculate T and T by summing up these terms orderby order, which is accurate but impractical for realistic devicesimulations. Alternatively, diagrammatic techniques will bedeveloped in the following sections to evaluate the summa-tion approximately. = ++ + + ... ij i k j i ≠ ≠≠ FIG. 2: Diagram representation of Eq.(15).
III. THE COHERENT POTENTIAL APPROXIMATION
In this section we present the formalism for calculatingtransmission fluctuation δT based on the CPA diagrammatictechnique. The main idea is to expand T and T into a se-ries of scattering terms each of which can be mapped into adiagram. At the CPA level, a subset of these diagrams (thenon-crossing diagrams) can be collected and summed up. Thediagrammatic technique was originally developed in Ref.13 tocalculate transport coefficients involving 3-Green’s functioncorrelators. Here, this technique is improved and generalizedto calculate δT involving 4-Green’s function correlators. A. The Γ -decomposition As shown in Eq.(4), transmission coefficient T is a traceof matrix product, and hence T is a product of two traceswhich is inconvenient to apply the diagrammatic technique.To proceed, we first rewrite T into a proper matrix prod-uct form. Using the Γ -decomposition technique introducedin Ref.19, the line-width function of the right electrode, Γ R ,can be decomposed as Γ R = (cid:80) n | W n (cid:105) (cid:104) W n | , where | W n (cid:105) is the n -th normalized eigenvector of the Γ R matrix . Con-sequently, using Eq.(4) T can be rewritten in the following Γ -decomposition form: T = ( Tr G r Γ L G a Γ R ) × ( Tr G r Γ L G a Γ R )= (cid:88) n Tr G r Γ L G a | W n (cid:105) (cid:104) W n | (cid:88) m Tr G r Γ L G a | W m (cid:105) (cid:104) W m | = (cid:88) n (cid:104) W n | G r Γ L G a | W n (cid:105) (cid:88) m (cid:104) W m | G r Γ L G a | W m (cid:105) = (cid:88) nm Tr G r Γ L G a X nm G r Γ L G a X † nm , (16)where X nm is defined as X nm ≡ | W n (cid:105) (cid:104) W m | .So the calculations of T and T are reduced tothe Green’s function correlators Tr G r X G a X andTr G r X G a X G r X G a X where X k is a definite quantitywhich is referred to as the vertex of the correlator. Notice that G r and G a always appear alternatively in T and T , as suchwe shall omit the superscripts r, a in the CPA diagrammaticexpansion without causing any ambiguity. B. The CPA diagrams
Eq.(16) indicates that we need to calculate various Green’sfunction correlators such as: I ≡ Tr GX GX , (17) I ≡ Tr GX GX GX , (18) I ≡ Tr GX GX GX GX . (19)To proceed we insert Eq.(15) into I n ( n = 2 , , ) to obtaina series expansion. In analogous to Eq.(15) and Fig.2, eachterm in the I n series expansion can be represented by a dia-gram: the thick line represents the full Green’s function G ;the thin line represents the unperturbed Green’s function G ;the blue dot represents the vertex X n ; the dotted line with ared dot represents the impurity scattering amplitude ˆ t iq . Thetrace operation is represented by a closed circle composed of G -lines and X -vertexes. If some impurity indices are identi-cal in the disorder average, the corresponding impurity linesneed to be contracted with each other. The major differencebetween the diagrams in this section and Fig.2 is that the for-mer diagrams represent terms after disorder average while thelatter represents terms before disorder average.Thus the lengthy series expansion of I n is nicely organizedinto a diagrammatic fashion. One can sum up the diagramsin a perturbative manner up to some finite order as done inRef.16. Alternatively, by selecting a subset of the diagrams,one can evaluate the diagrammatic summation to infinite or-der. In particular, the subset is called CPA diagrams selectedby the following two rules. (i) An impurity line on one G -line must contract with impurity line(s) of other G -line(s), andno dangling impurity line is allowed. The reason is that inCPA the partition of H and V is chosen such that t ri = t ai = 0 , (20)and hence diagrams with dangling impurity lines vanish (seeAppendix-A for details). (ii) Contracted impurity lines do notcross each other. Namely, only the non-crossing diagrams aretaken into account in the CPA diagrammatic summation. Inthe following subsections, CPA diagrams of I , I , and I will be analyzed in detail. = + + ... + +== FIG. 3: (color online) CPA diagrams of I . C. I diagrams By inserting Eq.(15) into Eq.(17) and applying CPA dia-gram rules, the I diagrams are obtained in Fig.3. In the first row of Fig.3, the diagram equation corresponds to the follow-ing algebraic equation:Tr GX GX = Tr G X G X + (cid:88) i q x i q Tr G ˆ t i q G X G ˆ t i q G X + (cid:88) i q (cid:88) i q i (cid:54) = i x i q x i q Tr G ˆ t i q G ˆ t i q G X G ˆ t i q G ˆ t i q G X + · · · . (21)The diagram representation in Fig.3 significantly simplifiesthe algebraic expression of Eq.(21).In the second and third rows of Fig.3, the diagrams of I is further simplified by using a bundled line (second row) anda dressed vertex (third row). The bundled line ˜ t (green thickline) is a collection of ladder diagrams. The vertex correction Λ is the combination of a bundled line ˜ t and a vertex X . Thedressed vertex Π (cyan shadow) is a vertex X plus its vertexcorrection Λ . The meaning of the diagram elements ˜ t , Λ and Π are explained in Fig.4.Given a vertex X , the corresponding vertex correction Λ issolved from the following equation: Λ i = (cid:88) q x iq t iq ( G XG ) ii t iq + (cid:88) j (cid:54) = i (cid:88) q x iq t iq ( G ) ij Λ j ( G ) ji t iq , (22)where Λ = diag ([Λ , Λ , · · · ]) is a diagonal matrix. Eq.(22)is derived by the recursive relation illustrated in Fig.5. Notethat Eq.(22) is identical to Eqs.(49,50) in Ref.12. = += + + + ...... += + FIG. 4: (color online) CPA diagram elements: the bundled line ˜ t , thevertex correction Λ , and the dressed vertex Π . D. I diagrams By inserting Eq.(15) into Eq.(18) and applying the CPA di-agram rules, I diagrams are obtained in Fig.6. In the firstthree rows of Fig.6, there are 16 diagrams constructed withbundled lines which are equivalent to Fig.3 of Ref.13. In thefourth row of Fig.6, the diagram number is reduced to two by = + FIG. 5: (color online) Diagram representation of Eq.(22). using the dressed vertex Π which has been defined in Fig.4. Inthe fifth row of Fig.6, the diagram number is reduced furtherto one by using the dressed vertex Π and the dressed doublevertex Π which is defined in Fig.7. E. I diagrams By inserting Eq.(15) into Eq.(19) and applying the CPAdiagram rules, I diagrams are obtained in Fig.8. There are256 diagrams if constructed only with the bundled lines (notshown). The diagram number is reduced to 16 if constructedwith bundled lines and dressed vertexes, as shown in the firstthree rows of Fig.8. The diagram number is reduced to 6 ifconstructed with bundled lines, dressed vertexes and dresseddouble vertexes, as shown in the fourth row of Fig.8. It is clearthat the using of the dressed vertex and dressed double vertexgreatly reduces the number of CPA diagrams. F. The sum rules
How do we know that all the CPA diagrams have beenincluded in the diagrammatic summation? There are someWard’s type identities in Green’s functions which are helpfulto verify the completeness of the CPA diagrams. The identi-ties reduce a product of Green’s functions to products of fewerGreen’s functions. By applying disorder average to both sidesof the identity, the identity must remain valid if the average isdone rigorously. This way the higher level correlators (e.g. 4-Green’s function correlators) are related to lower level corre-lators (e.g., 2-Green’s function correlators). The amazing fea-ture of CPA is that the identity still holds even if approxima-tions are made on both sides of the identity. In this sense, CPAis a consistent approximation for the Green’s function corre-lators. These identities can thus be used to verify theoreticalderivations as well as numerical implementations. Missing asingle diagram will make the identities unbalanced.In particular, the identities for testing I , I , and I arelisted below: G r Σ ra G a = G r − G a , (23) G r Σ ra G a Σ ra G r = G r Σ ra G r + G a − G r , (24) G r Σ ra G a Σ ra G r Σ ra G a = G r Σ ra G r + G a Σ ra G a − G r − G a ) , (25) where Σ ra ≡ Σ r − Σ a . Note that in these equalities, the lefthand side involves higher level correlator while the right handside involve lower level correlators. Our analytical formalismand numerical computation have been verified by confirmingthe equality to high precision. In Appendix-B, we provide ananalytical proof of the identity Eq.(23). G. Summary of CPA diagram technique
In this section, CPA diagrams for evaluating Green’s func-tion correlators I , I , I are presented. I and I have beeninvestigated in Ref.13 and are included here for completenessand improvement. For the first time in literature, we have de-rived the CPA diagrams for I and reduced the diagram num-ber from 256 to 6 by using dressed vertex and dressed doublevertex.By using CPA diagrams of I and I , transmission fluc-tuation δT = (cid:113) T − T can be calculated as follows: (i)Calculate G r , t riq , G a , t aiq by solving CPA condition Eq.(20).The details are presented in Appendix-A; (ii) Calculate T byusing the disorder average of Eq.(16) and the CPA diagramsin Fig.8; (iii) Calculate T by using the disorder average ofEq.(4) and the CPA diagrams in Fig.3. The dressed vertexesin the CPA diagrams can be calculated by using Eq.(22). It isconcluded that the RDF induced transmission fluctuation canbe calculated by the CPA diagrammatic technique presentedin this section. IV. THE LOW CONCENTRATION APPROXIMATION
In the last section we have presented a general formal-ism based on the CPA diagrammatic technique to evaluatethe transmission fluctuation δT . It is general in the sensethat RDF is calculated for arbitrary impurity concentration x .Nevertheless, in semiconductor devices (e.g. transistors) thedoping concentration is always very low. Even for heavilydoped Si at a doping level cm − , the impurity concentra-tion amounts to x ∼ × − which is a small parameter.Therefore one can carry out a perturbative expansion to thelowest order of the small parameter x to evaluate δT , whichis referred to as the low concentration approximation (LCA).This is especially useful for analyzing RDF induced device-to-device variability in semiconductor nanoelectronics. Thissection is devoted to present the LCA formalism.Let q = 0 represent the host material atom specie and q > impurity atom species. Low concentration means that theconcentration of host material atom is much larger than thatof impurity atoms, i.e., x i,q =0 (cid:29) x i,q> . The main idea ofLCA is to collect the lowest order terms in δT which areproportional to x i,q> . Because the impurity concentration issmall, in the partition of the total Hamiltonian Eqs.(7,8), wenaturally choose H to be the Hamiltonian of the host materialand V to be the difference between impurity atoms and hostatoms. Consequently the disorder scattering potential V iq is V iq = ε iq − ε i , (26) = + + ++++ + + +++ + + ++== + FIG. 6: (color online) CPA diagrams of I . = + FIG. 7: (color online) CPA diagram element: dressed double vertex Π . where ε iq is the on-site energy of impurity atom and ε i is theon-site energy of host atom. This is in contract to the CPAdiagrammatic formalism of the last section in which H and V have been chosen such that the CPA condition t ri = t ai = 0 is satisfied.The simplicity of LCA is that it does not need Γ -decomposition as in CPA. One can directly substitute Eq.(15)and its advanced counterpart into Eq.(4) and its square to ob-tain a series expansion for T and T . Averaging over disorderconfigurations and collecting the terms up to the first order of x i, ˙ q> , T and T can be obtained and represented by the LCAdiagrams in Fig.9 and Fig.10, respectively.The meaning of LCA diagrams is similar to that of CPAdiagrams: The thin line represents unperturbed Green’s func-tion G r or G a ; The blue dot represents vertex Γ L or Γ R ; Thedotted line with a red dot represents impurity scattering am-plitude ˆ t ri or ˆ t ai . The closed Green’s function circle means tocarry out trace operation. The contraction of impurity linesmeans that the disorder site indices are the same. To be spe-cific, the LCA diagrams (1), (2), (3), (4) in Fig.9 correspondto the following algebraic expressions in order:Tr ( G r Γ L G a Γ R ) , (cid:88) iq x iq Tr (cid:0) G r Γ L G a ˆ t aiq G a Γ R (cid:1) , (cid:88) iq x iq Tr (cid:0) G r ˆ t riq G r Γ L G a Γ R (cid:1) , (cid:88) iq x iq Tr (cid:0) G r ˆ t riq G r Γ L G a ˆ t aiq G a Γ R (cid:1) . The LCA diagrams (1), (6), (11), (16) in Fig.10 correspond tothe following algebraic expressions in order:Tr ( G r Γ L G a Γ R ) Tr ( G r Γ L G a Γ R ) , (cid:88) iq x iq Tr (cid:0) G r ˆ t riq G r Γ L G a ˆ t aiq G a Γ R (cid:1) Tr ( G r Γ L G a Γ R ) , (cid:88) iq x iq Tr (cid:0) G r ˆ t riq G r Γ L G a Γ R (cid:1) Tr (cid:0) G r ˆ t riq G r Γ L G a Γ R (cid:1) , (cid:88) iq x iq Tr (cid:0) G r ˆ t riq G r Γ L G a ˆ t aiq G a Γ R (cid:1) Tr (cid:0) G r ˆ t riq G r Γ L G a ˆ t aiq G a Γ R (cid:1) . Of the 16 LCA diagrams for T , 7 diagrams (from (1) to (7)in Fig.10) are unconnected and will cancel with the 7 LCA di-agrams from T in calculating δT . The summation of theremaining 9 diagrams (from (8) to (16) in Fig.10) can be fur-ther simplified as δT = (cid:88) i,q> x iq (cid:16) Y αiq + Y βiq + Y γiq (cid:17) , (27) = + +++ ++++ ++ - += +++ ++++ + FIG. 8: (color online) CPA diagrams of I . ⑴ r G a G L R ⑵ r G a G L R ⑶ r G a G L R ⑷ r G a G L R FIG. 9: (color online) LCA diagrams of T . where Y αiq = Tr (cid:8) t aiq [ G a Γ R G r Γ L G a ] ii (cid:9) , (28) Y βiq = Tr (cid:8) t riq [ G r Γ L G a Γ R G r ] ii (cid:9) , (29) Y γiq = Tr (cid:8) t riq [ G r Γ L G a ] ii t aiq [ G a Γ R G r ] ii (cid:9) , (30)in which (cid:0) Y αiq (cid:1) ∗ = Y βiq and (cid:0) Y γiq (cid:1) ∗ = Y γiq . It follows that δT > which is consistent with the physical meaning ofthis quantity. Note that the summation over i and q in Eq.(27)clearly identifies the contribution of each impurity specie and disorder site to the total transmission fluctuation. Eq.(27) to-gether with the definition of G r in Eq.(10), t riq in Eq.(14), and V iq in Eq.(26) are the central results of this section. V. FORMULATION IN FOURIER SPACE
Having presented two theoretical methods for computing δT , i.e. the CPA diagrammatic formalism and the LCA dia-grammatic formalism, we now consider an important specialsituation where two-probe systems are “periodic” in the trans-verse dimensions. When there is no disorder, periodicity iswell defined, and one can identify a unitcell in the transversedimensions and apply the Bloch theorem by Fourier trans-form. In disordered two-probe systems, one can also iden-tify a unitcell but the situation is more complicated. On theone hand, the Hamiltonian does not have translational sym-metry in the presence of random disorder thus Bloch theorembreaks down. On the other hand, the disorder averaged physi-cal quantities are still periodic and can be Fourier transformed.The formalisms developed in the previous two sections needto be modified slightly to adapt to such disordered “periodic”two-probe systems.Assume that a disordered two-probe system has periodicityin one transverse dimension. Define the dimensionless crystalmomentum k as k = k · a where k is the wave vector and a is the unitcell vector of the periodic dimension . A periodicphysical quantity Y as a function of unitcell indices I and I should be only dependent on the index difference I − I .Therefore Y I ≡ Y I − I can be transformed into the Fourierspace Y ( k ) = (cid:88) I e − i kI Y I , r G a G L R r G a G L R ⑴ r G a G L R r G a G L R ⑵ r G a G L R r G a G L R ⑶ r G a G L R r G a G L R ⑷ r G a G L R r G a G L R ⑸ r G a G L R r G a G L R ⑹ r G a G L R r G a G L R ⑺ r G a G L R r G a G L R ⑻ r G a G L R r G a G L R ⑼ r G a G L R r G a G L R ⑽ r G a G L R r G a G L R ⑾ r G a G L R r G a G L R ⑿ r G a G L R r G a G L R ⒀ r G a G L R r G a G L R ⒁ r G a G L R r G a G L R ⒂ r G a G L R r G a G L R ⒃ FIG. 10: (color online) LCA diagrams of T . For example, H and Σ rβ do not contain randomness and canbe Fourier transformed into H ( k ) and Σ rβ ( k ) . Consequently G r ( k ) = [ E − H ( k ) − Σ r ( k )] − , Γ β ( k ) = i (cid:2) Σ rβ ( k ) − Σ aβ ( k ) (cid:3) . To obtain the on-site quantity Y ii , one needs to integrate over k (inverse Fourier transform) Y ii = (cid:90) + π − π dk π Y ii ( k ) . To carry out Γ -decomposition, the summation over the elec-trode conducting channel n should be replaced by an integralover k in addition to the summation over n , i.e., (cid:80) n −→ (cid:82) + π − π dk π (cid:80) n . The necessary modifications of CPA formalismand LCA formalism are presented explicitly as follows.For the CPA diagrammatic formalism presented in SectionIII, the Γ -decomposition Eq.(16) should be modified as: T = (cid:90) + π − π dk π (cid:90) + π − π dk (cid:48) π (cid:88) nn (cid:48) Tr G r ( k ) Γ L ( k ) G a ( k ) X nk,n (cid:48) k (cid:48) G r ( k (cid:48) ) Γ L ( k (cid:48) ) G a ( k (cid:48) ) X † nk,n (cid:48) k (cid:48) , (31)0where X nk,n (cid:48) k (cid:48) is defined as X nk,n (cid:48) k (cid:48) ≡ | W n ( k ) (cid:105) (cid:104) W n (cid:48) ( k (cid:48) ) | , in which the eigenvector | W n ( k ) (cid:105) comes from the k -dependent Γ -decomposition of Γ R ( k )Γ R ( k ) = (cid:88) n | W n ( k ) (cid:105) (cid:104) W n ( k ) | . Moreover, the vertex correction Eq.(22) needs to be modifiedas: Λ i = (cid:88) q x iq t iq (cid:26)(cid:90) + π − π dk π [ G ( k ) X ( k ) G ( k )] ii (cid:27) t iq + (cid:88) q x iq t iq (cid:26)(cid:90) + π − π dk π [ G ( k ) Λ G ( k )] ii (cid:27) t iq − (cid:88) q x iq t iq (cid:20)(cid:90) + π − π dk π [ G ( k )] ii (cid:21) Λ i (cid:20)(cid:90) + π − π dk π [ G ( k )] ii (cid:21) t iq , (32)in which X ( k ) is the Fourier transform of X .For the LCA diagrammatic formalism presented in Section IV, Y αiq , Y βiq , Y γiq , and t riq in Eq.(27) should be modified as: Y αiq = Tr (cid:26) t aiq (cid:20)(cid:90) + π − π dk π G a ( k ) Γ R ( k ) G r ( k ) Γ L ( k ) G a ( k ) (cid:21) ii (cid:27) , (33) Y βiq = Tr (cid:26) t riq (cid:20)(cid:90) + π − π dk π G r ( k ) Γ L ( k ) G a ( k ) Γ R ( k ) G r ( k ) (cid:21) ii (cid:27) , (34) Y γiq = Tr (cid:26) t riq (cid:20)(cid:90) + π − π dk π G r ( k ) Γ L ( k ) G a ( k ) (cid:21) ii t aiq (cid:20)(cid:90) + π − π dk (cid:48) π G a ( k (cid:48) ) Γ R ( k (cid:48) ) G r ( k (cid:48) ) (cid:21) ii (cid:27) . (35)and t riq = (cid:104) ( ε iq − ε i ) − − G r ,ii (cid:105) − ( q > , (36)where G r ,ii is obtained as G r ,ii = (cid:90) + π − π dk π [ G r ( k )] ii . VI. FURTHER DISCUSSIONS
Several important issues are worth further discussions in-cluding the scaling behavior of the transmission fluctuation δT , the comparison of CPA and LCA diagrammatic for-malisms, the generalization of CPA and LCA to atomic mod-els of nanoelectronics, the application of CPA and LCA tocompute other physical quantities, and the procedure to deter-mine the variation of threshold voltage for field effect transis-tors. A. Scaling
In two-probe systems with transverse periodicity, transmis-sion coefficient T and transmission fluctuation δT are calcu-lated for a single unitcell in the transverse dimensions, as dis-cussed in Section V. It should be emphasized that T and δT have very different scaling behaviors with respect to the crosssection area. Suppose a cross section contains N unitcells inthe transversion dimensions, transmission is proportional to N but transmission fluctuation is proportional to √N . In thelimit of infinitely large transverse cross section, the ratio of δT over T goes to zero which is the thermodynamic limit. It istherefore clear that the device variability due to RDF is mostsignificant in nano-scale systems whose cross section area isnot sufficient large to exhibit self-averaging of the disorderconfigurations. B. CPA vs LCA
We have so far presented two diagrammatic formalisms,CPA and LCA, for calculating δT . A comparison of CPA1and LCA is as follows. (i) In principle CPA is more accu-rate than LCA, because from the diagram point of view LCAonly considers the lowest order diagrams while CPA consid-ers all non-crossing diagrams to infinite order. As a resultLCA is applicable to the low concentration limit while CPA isapplicable to a wider concentration range. Numerically weshall compare the two methods in Section VII. (ii) To ap-ply CPA formalism to calculate δT , one has to carry out Γ -decomposition to rewrite T into a proper matrix product form(see Eq.(16)). In contrast, the LCA formalism does not require Γ -decomposition and can be applied directly to calculate δT .The Γ -decomposition leads to double summation and double k -integral (see Eq.(31)) over conducting channels of the elec-trode and significantly increase the computational cost. (iii)CPA is far more complicated to implement than LCA, becausethe former needs to solve the CPA equations as well as sev-eral vertex correction equations iteratively as discussed at theend of Section III-G. In contrast, LCA provides an explicitformula, Eq.(27), to calculate the transmission fluctuation di-rectly. (iv) To reduce the computational cost in modeling na-noelectronic devices, it is often desirable to partition a two-probe system into many slices along the transport directionand apply a numerical trick – the principal layer algorithm,in the Green’s function’s calculation . This very useful algo-rithm can be easily integrated into the LCA formulism but itis incompatible with the CPA diagrams. In short, the CPA di-agrammatic formalism is much more complicated and costlythan LCA to calculate δT due to the reasons listed in (ii) to(iv), although CPA is more accurate and applicable to a widerconcentration range. C. Generalization to atomic model
It is straightforward to generalize both CPA and LCA for-malisms to the atomic model of nanoelectronic devices. As-sume that each atom is represented by M atomic orbitals, theon-site energy ε iq should be replaced by an M × M matrixblock. Correspondingly, the variable V iq , t riq , G r ,ii , ˜ ε riq , Λ i also become M × M matrix blocks. Meanwhile the formu-lation should be adapted according to the definition of theGreen’s functions in the specific method.For example, in the first principle model implementing lin-ear muffin-tin orbital (LMTO) method , the on-site energy ε iq should be replaced by the potential function − P iq ( E ) which is a ( L max + 1) × ( L max + 1) diagonal matrix blockwhere L max is the maximum angular momentum quantumnumber. The coherent potential ˜ ε ri should be replaced by theLMTO coherent potential − ˜ P ri ( E ) which is a ( L max + 1) × ( L max + 1) full matrix block. Moreover, the definition ofauxiliary Green’s function in the LMTO method is very differ-ent from that of standard Green’s function presented in Sec-tion III and IV, and hence the formulation need to be modi-fied accordingly. In the CPA formalism, E − T − ˜ ε r in thefifth row of Eq.(A1) needs to be replaced by ˜ P r ( E ) − S ( k ) where ˜ P r ( E ) is the LMTO coherent potential and S ( k ) is theFourier transformed structure constant. In the LCA formal-ism, E − H in Eq.(10) should be replaced by P ( E ) − S ( k ) , where P ( E ) is the potential function of the host material.For technical details of LMTO method, we refer interestedreaders to the monographs of Ref.23–25.This way, we have implemented a transmission fluctuationanalyzer based on the LCA diagrammatic formalism and theLMTO method in the first principle nano-scale device simula-tion package NanoDsim , which will be applied in an exam-ple in Section VII. D. Other physical quantities
In the NEGF approach, to calculate a physical quantity,the general idea is to first express the quantity in terms ofGreen’s functions and then evaluate these Green’s functions.In Ref.27, disorder averaged Green’s functions G r and G < have been solved from the equations of nonequilibrium coher-ent potential approximation (NECPA). Therefore if a quantitycan be expressed as a linear combination of G r and G < , thedisorder average of this quantity can be readily calculated withNECPA. It has been shown in Ref.27 that electric current andoccupation number belong to this category.Some physical quantities, however, involve Green’s func-tion correlators which are beyond the scope of NECPA. CPAand LCA formalisms presented in this work can systemati-cally calculate disorder averaged Green’s function correlatorsand related physical quantities. In addition to transmissionfluctuations studied here, CPA and LCA techniques can alsobe applied to investigate other quantities. For example, theshot noise can be expressed as S = Tr (cid:104) G r Γ L G a Γ R − ( G r Γ L G a Γ R ) (cid:105) , and the disorder averaged shot noise S can be readily evalu-ated with CPA or LCA formalism. E. Variation of the threshold voltage
For field effect transistors it is relevant to predict the vari-ation of threshold voltage in addition to the variations of on-state and off-state current due to RDF. This can be done withthe following procedure. (i) Calculate the disorder averagedcurrent as a function of gate voltage I = F ( V g ) ; (ii) De-termine the averaged threshold voltage V T from F ( V g ) ; (iii)Calculate the current fluctuation δI at V T by using the CPAor LCA formalism of this work; (iv) Estimate the variation ofthe threshold voltage by the slope of F ( V g ) at V T : δV T ≈ δI (cid:12)(cid:12) F (cid:48) (cid:0) V T (cid:1)(cid:12)(cid:12) . VII. NUMERICAL EXAMPLES
In this section, CPA and LCA formalisms are applied totight-binding (TB) models and an atomic model to investigate2transmission fluctuation induced by RDF. Three examples areprovided: a TB model with finite cross section, a TB modelwith periodic transverse cross section, and an atomic modelwith periodic transverse cross section.
A. Tight binding model: finite cross section
This example investigates transmission fluctuation in a onedimensional (1D) tight-binding nano-ribbon. The system isshown in the inset of Fig.11a where the yellow sites repre-sent host sites whose on-site energies are set to zero. The redsites represent impurity sites whose on-site energies are eitherzero with the probability − x or . with the probability x .Only the nearest neighbors have interactions with a couplingstrength set to unity. Fig.11a also shows transmission coeffi-cient T ( E ) in the clean limit ( x = 0 ). As expected, T ( E ) isan integer step-like curve which coincides with the number ofthe conducting channels at the energy E .For this simple example the exact solution is available bybrute force enumeration. Namely, T ( E ) can be calculatedfor all disorder configurations and δT can be evaluated ex-actly. This example sets a benchmark to check the validityand accuracy of CPA and LCA. In Fig.11b to Fig.11h, δT iscalculated by using three different methods: exact, LCA, andCPA. The disorder concentration is increased systematicallyfrom x = 0 . to x = 0 . .A few observations are in order. (1) For x (cid:54) . , bothLCA and CPA give very satisfactory results in comparison tothe exact solution. For x (cid:62) . , both LCA and CPA solutionbecome less accurate. The reason is that LCA neglects higherorder terms of concentration x while CPA neglects crossingdiagrams. (2) LCA solution is always physical in the senseof δT > which is actually expected from Eq.(27). CPAsolution, however, may give non-physical results in some en-ergies where δT < . In Fig.11, the non-physical pointshave been reset to δT = 0 . (3) Large transmission fluctua-tion occurs at energies where the transmission channel num-ber changes drastically. It implies that current fluctuation canbe suppressed if the bias voltage window is tuned to locate inan energy plateau with slow varying conducting channel num-ber. B. Tight binding model: periodic cross section
This example investigates transmission fluctuation in a twodimensional (2D) tight-binding lattice. The system is shownin the inset of Fig.12a where the yellow sites represent hostatoms whose on-site energies are set to zero; the red sites rep-resent impurities whose on-site energies are either zero withthe probability − x or . with the probability x . Only thenearest neighbors have interactions with a coupling strengthset to unity. Fig.12a also shows the transmission coefficient T ( E ) in the clean limit ( x = 0 ). T ( E ) is has a sharp peakat E = 0 which can be well understood by the correspondingband structure of this lattice. For this example exact solution is unavailable due to theinfinite degrees of freedom. The CPA solution is very expen-sive due to double summation and double k -integral in the Γ -decomposition Eq.(31). Since the LCA solution of finite crosssection has been checked in the previous subsection, the LCAsolution of periodic cross section will be checked against it byusing a large finite cross section containing rows.For this 2D model, the LCA solution of the periodic crosssection agrees very well with that of large finite cross section,as expected. Note that the solution for the finite cross sectionmodel must be re-scaled with a proper scaling factor √ as discussed in subsection VI-A. The Transmission fluctuationshows a sharp peak around E = 0 where the transmission alsohas a spike. An impression is that the transmission fluctuationis more pronounced in the energy regime where the transmis-sion coefficient changes rapidly. C. Atomic model: periodic cross section
This example investigates the transmission fluctuation in athree-dimensional (3D) Cu lattice having random atomicvacancies by using an atomic implementation of the LCA for-malism. The system has a periodic cross section and the trans-port is perpendicular to the Cu (111) direction. In the atomicmodel, the left and right semi-infinite Cu electrodes are con-nected to a central region which consists of 5 perfect Cu layers(buffer layer), 15 disordered Cu layers in the alloy model ofCu . Vac . (“Vac” indicates vacancy), and another 5 per-fect Cu layers (buffer layer). Namely, the central region canbe represented by the formula [Cu] -[Cu . Vac . ] -[Cu] .The calculation proceeds in two steps. First, we self-consistently solve the device Hamiltonian of the open two-probe system using the NECPA-LMTO method as imple-mented in the NanoDsim package . Second, we calculatethe transmission fluctuation using the LCA formalism com-bined with the LMTO method which has been implementedinto the NanoDsim package as a post-analysis tool. The resultfor [Cu] -[Cu . Vac . ] -[Cu] is presented in Fig.13.The transmission fluctuation exhibits a strong energy de-pendence. This is quite interesting since it means the con-ductance fluctuation can be effectively suppressed by shiftingthe Fermi energy. In the vicinity of E = − . eV , δT israther small although T changes rapidly, which seems to bedifferent than the observations in the tight-binding examples.Further analysis shows that the density of states is dominatedby d -wave in the energy regime around E = − . eV andis a mixtures of s -wave, p -wave and d -wave well above thisenergy. Transmission fluctuation is enhanced due to disorderscattering among different angular momentum states. This in-dicates that the transmission fluctuation is not only affectedby the number of the conducting channels (as in the TB mod-els) but also by the angular momentum states of the channelswhen realistic atomic models are considered.3 -4 -2 0 2 40.000.010.020.030.04-4 -2 0 2 40.000.040.080.12 -4 -2 0 2 40.00.10.20.30.4-4 -2 0 2 40.00.20.40.6 -4 -2 0 2 40.00.20.40.6-4 -2 0 2 40.00.20.40.60.8 -4 -2 0 2 40.00.20.40.60.8-4 -2 0 2 40123 (a) x = 0 exact LCA CPA E T EE T T E T T E T E T E T (b) x = 0.001 exact LCA CPA (c) x = 0.01 exact LCA CPA E (d) x = 0.1 exact LCA CPA (e) x = 0.2 exact LCA CPA (f) x = 0.3 exact LCA CPA (g) x = 0.4 exact LCA CPA (h) x = 0.5 FIG. 11: (color online) Transmission fluctuation in the tight-binding model with finite cross section shown in the inset of (a). (a) Transmission T ( E ) in the clean limit. (b) to (h), Transmission fluctuation δT ( E ) at different doping concentrations x . For comparison, δT is calculatedwith three methods: exact, LCA, and CPA. VIII. CONCLUSION
In this work, we have developed two theoretical formalismsbased on CPA and LCA to predict device-to-device variabil-ity induced by random dopant fluctuation. The advantage of our theory is that statistical averaging due to RDF is carriedout analytically to avoid large number of dopant configurationsampling in device simulations.The numerical accuracy of CPA and LCA formalism de-pends on the doping concentration x . For x (cid:54) . , both4 -4 -2 0 2 40.0000.0050.0100.0150.0200.025-4 -2 0 2 40.00.20.40.60.81.0 (a) x = 0 T LCA-finite LCA-periodic E E T (b) x = 0.001 FIG. 12: (color online) Transmission fluctuation in the tight-bindingmodel with periodic cross section shown in the inset of (a). (a) Trans-mission T ( E ) in the clean limit. (b) Transmission fluctuation δT ( E ) for the doping concentration x = 0 . . For comparison, δT is cal-culated with two methods: LCA-finite and LCA-periodic. CPA and LCA solutions are satisfactory, as shown in the com-parison to the exact solution of 1D TB model. For x (cid:62) . ,both CPA and LCA become numerically less accurate eventhough they still capture a rough trend of the transmissionfluctuation as demonstrated by the 1D TB model. In LCAwe have neglected high order terms in the x -expansion, whilein CPA we have neglected the crossing diagrams. These ap-proximations limit the accuracy of the theory to the relativelylow impurity concentrations. We note that for essentially allthe practical semiconductor devices, the dopant concentra-tion is well within the applicability range of our formalisms.In numerical modeling, the LCA is easier and perhaps morepractical for realistic nanoelectronic devices because an ex-plicit formula Eq.(27) is available and the computational costis much cheaper than that of CPA. We have also implementedthe LCA theory into the first principles device modeling pack-age NanoDsim so that first principles analysis of device-to-device variability can now be carried out without any phe-nomenological parameters.Preliminary studies indicate that transmission fluctuation ismost pronounced in the energy regime where the number ofthe conducting channels varies rapidly. In addition, angularmomentum states of the conducting channels also play an es-sential role. Since the fluctuation strongly depends on theelectron energy, our numerical simulation suggests that the -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00400800120016002000-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0 (a) E (eV) T total s-wave p-wave d-wave E (eV) D O S (b) FIG. 13: (color online) Transmission fluctuation of the 3D Cu two-probe lattice with random vacancy defects. (a) Transmissionfluctuation δT on top of Transmission T versus energy E . The areaof the unitcell cross section is 5.64 ˚ A . (b) Total and angular mo-mentum resolved density of states versus E . RDF induced transmission fluctuation could be suppressedby engineering the bias voltage window to a proper energyregime. Finally, we have so far focused on investigating theRDF induced transmission fluctuation in nanostructures, ourtheory and numerical implementation can be applied to studymany other physical quantities such as the shot noise, the fluc-tuation of threshold voltage, as well as the device variability inspintronics. We hope to report these and other investigationsin future publications.
ACKNOWLEDGEMENT
We wish to thank Dr. Jianing Zhuang and Prof. Jian Wangfor valuable discussions concerning their work in Ref.16. Wethank Dr. Ferdows Zahid for bringing our attention to Ref.5and discussions on practical device issues of RDF.
Appendix A: The CPA condition
In this appendix, we present how to calculate quantities G r , t riq , G a , t aiq by using the CPA condition Eq.(20). As mentionedin Section II, there are some freedom to partition H into H V , i.e. Eqs.(7,8). CPA takes the advantage of this freedomand chooses a special partition such that the disorder averagedscattering vanishes, i.e. Eq.(20).Assume that the Hamiltonian matrix is written as H = T + ε where T is the off-diagonal part of the Hamiltonian and ε the diagonal part. T is a definite matrix and does not haveany randomness. In contrast, the diagonal matrix ε containsdiscrete random variables, the i -th diagonal element ε i cantake the value ε iq with the probability x iq and (cid:80) q x iq = 1 .One can introduce a diagonal quantity called coherent poten-tial ˜ ε r ≡ diag ([˜ ε r , ˜ ε r , · · · ]) and define H and V as H = T + ˜ ε r ,V = ε − ˜ ε r . By imposing CPA condition Eq.(20) to the above partition of H and V , ˜ ε r can be solved from the following CPA equa-tions: t ri = (cid:80) q x iq t riq = 0 ,t riq = V iq (cid:2) − G ri V iq (cid:3) − ,V iq = ε iq − ˜ ε ri ,G ri = (cid:2) G r (cid:3) ii ,G r = ( E − T − ˜ ε r − Σ r ) − . (A1)Once ˜ ε r is solved, G r = G r and t riq are also known fromEq.(A1). Finally, G a and t aiq are simply Hermitian conjugatesof G r and t riq , respectively. Appendix B: Proof of Eq.(23)
In this appendix, we provide an analytical proof of Eq.(23).By using the vertex correction, the left hand side of Eq.(23)can be obtained as G r Σ ra G a = G r (Σ ra + Λ) G a , (B1)where Λ is the vertex correction determined by Eq.(22) with X = Σ ra . By using the expressions of G r and G a in CPA,the right hand side of Eq.(23) can be transformed into a similarform as the left hand side: G r − G a = G r (cid:16) Σ ra + ˜Λ (cid:17) G a (B2) where ˜Λ ≡ ˜ ε r − ˜ ε a . G r and G a are determined by the coherentpotential ˜ ε r and ˜ ε a (see Eq.(A1)), G r = ( E − H − ˜ ε r − Σ r ) − ,G a = ( E − H − ˜ ε a − Σ a ) − . Comparing Eq.(B1) and Eq.(B2), it is inferred that ˜Λ and Λ must be identical. Also note that the vertex correction Eq.(22)for Λ is an inhomogeneous linear equation thus has a uniquesolution. Hence the identity is proved if and only if ˜Λ satisfiesEq.(22).By using CPA condition Eq.(A1) and its Hermitian conju-gate t riq = (cid:104) ( V iq − ˜ ε r ) − − G ri (cid:105) − ,t aiq = (cid:104) ( V iq − ˜ ε a ) − − G ai (cid:105) − , one can derive the equation for ˜Λ by eliminating V iq (cid:0) t a − iq + G ai (cid:1) − − (cid:0) t r − iq + G ri (cid:1) − = ˜Λ . (B3)After some algebra, the equation of ˜Λ can be simplified as t aiq − t riq + t riq ( G ri − G ai ) t aiq = (cid:0) t riq G ri (cid:1) ˜Λ i (cid:0) G ai t aiq (cid:1) . (B4)By using Eq.(B2), it is obtained t aiq − t riq + t riq (cid:104) G r (cid:16) Σ ra + ˜Λ (cid:17) G a (cid:105) ii t aiq = (cid:0) t riq G ri (cid:1) ˜Λ i (cid:0) G ai t aiq (cid:1) . (B5)Notice that (cid:80) q x iq = 1 due to normalization, (cid:80) q x iq t riq =0 and (cid:80) q x iq t aiq = 0 due to the CPA condition. Applyingthe weighed summation (cid:80) q x iq on both sides of Eq.(B5), it isderived: (cid:88) q x iq t riq (cid:104) G r (cid:16) Σ ra + ˜Λ (cid:17) G a (cid:105) ii t aiq = ˜Λ i + (cid:88) q x iq t riq G ri ˜Λ i G ai t aiq , (B6)which is equivalent to Eq.(22) and thus proves Eq.(23). ∗ Electronic address: [email protected] International technology roadmap for semiconductors (2009), http://public.itrs.net/ . A. Asenov, IEEE Trans. Electron Deveces , 2505 (1998). A. Brown, A. Asenov and J. Watling, IEEE Transactions on Nan-otechnology , 195 (2002). R. Wang et al , IEEE Trans. Electron Deveces , 2317 (2011). See, for example, the
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