Determination of intrinsic lifetime of edge magnetoplasmons
Ken-ichi Sasaki, Shuichi Murakami, Yasuhiro Tokura, Hideki Yamamoto
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Determination of intrinsic lifetime of edge magnetoplasmons
Ken-ichi Sasaki ∗ NTT Basic Research Laboratories, NTT Corporation,3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan
Shuichi Murakami
Department of Physics, Tokyo Institute of Technology,2-12-1 Ookayama, Meguro, Tokyo 152-8551, Japan
Yasuhiro Tokura
NTT Basic Research Laboratories, NTT Corporation,3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan andFaculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
Hideki Yamamoto
NTT Basic Research Laboratories, NTT Corporation,3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan (Dated: September 5, 2018)It is known that peculiar plasmons whose frequencies are purely imaginary exist in the interior ofa two-dimensional electronic system described by the Drude model. We show that when an externalmagnetic field is applied to the system, these bulk plasmons are still non-oscillating and are isolatedfrom the magnetoplasmons by the energy gap of the cyclotron frequency. These are mainly in atransverse magnetic mode and can combine with a transverse electronic mode locally at an edgeof the system to form edge magnetoplasmons. With this observation, we reveal the intrinsic longlifetime of edge magnetoplasmons for the first time.
Many types of intriguing phenomena can emerge at theedge of a material that are invisible or hiding in the in-terior for some reason. Edge magnetoplasmon is such anexample; it is a gapless collective excitation that appearsat the edge of a two-dimensional electron gas (2DEG) un-der the application of an external magnetic field.
Theproperties unique to the edge magnetoplasmons, such asthe localization length and dispersion relation, were cal-culated by Volkov and Mikhailov. They succeeded insolving an integral equation of the electric potential us-ing the Wiener-Hopf method.
Meanwhile, an internalmagnetic field, which is coupled to the potential throughMaxwell equations, is neglected, and this simplificationprevents the lifetime of the edge magnetoplasmon ( τ ∗ inRef. 9) from being determined and also obscures the mag-netic configurations of the excitations. The fact that thelocalization length and dispersion relation are dependenton the lifetime makes it difficult to analyze experimentalresults.In this paper, we determine the intrinsic lifetime ofthe edge magnetoplasmon. Our analyses are based ontwo observations. The first is that there is a purely re-laxational state with a very long lifetime in the interiorof a 2DEG. The second is that the state acquires a non-zero real part of the frequency through localization andstarts to propagate. By showing that the properties ofthe localized state are consistent with those of the edgemagnetoplasmons, we identified the purely relaxationalstate as the bulk counterpart of the edge magnetoplas-mons and determined the lifetime of the edge magneto-plasmons. Our results show that the internal magnetic field normal to the layer is strongly suppressed in the in-terior, which partly justifies the assumption used in thepast and may lead us to a more complete description ofthe edge magnetoplasmons.We begin by reviewing a mathematical treatment ofplasmons. The fact that magnetic fields are discontinu-ous at a 2DEG layer plays a central role in the forma-tion of localized surface plasmons. The plasmons areclassified into transverse magnetic (TM) and transverseelectric (TE) modes with respect to their eigenvectors, asshown in Fig. 1. The electric fields of these eigenmodesare written in terms of the localization length in the di-rection normal to the layer α − , in-plane wavevector k y ,and angular frequency ω as E x = E TE x e i ( k y y − ωt ) e − α | z | ,E y = E TM y e i ( k y y − ωt ) e − α | z | ,E z = E TM z e i ( k y y − ωt ) e − α | z | , (1)where E TE x ( E TM y and E TM z ) is the amplitude of the TE(TM) mode. Note that the E i values in Eq. (1) are pro-portional to e − αz ( e + αz ) for z > z < z < E z with − E z becauseGauss’s law for free space, ∇· E = 0, must be satisfied forboth z > z <
0. The magnetic fields are obtainedfrom Eq. (1) using Faraday’s law, ∇ × E = − ∂ B /∂t . For z >
0, we have iωB x = ik y E z + αE y ,iωB y = − αE x ,iωB z = − ik y E x . (2)For z <
0, we replace α and E z on the right-hand side ofEq. (2) with − α and − E z , respectively. As a result, B x and B y are discontinuous at z = 0 as shown in Fig. 1.By applying Stokes’ theorem to Amp´ere’s circuital law ofthe Maxwell equations, c ∇ × B = ε∂ E /∂t + j /ǫ where j = ( j x , j y , δ ( z ) is the electronic current flowing in alayer, we find that the discontinuity of B x is related to j y as c ( B x ( z = 0+) − B x ( z = 0 − )) = j y /ǫ . Becauseof Ohm’s law, j (on the right-hand side) is proportionalto the in-plane electric fields as j i = σ ix ( ω ) E x | z =0 + σ iy ( ω ) E y | z =0 with the coefficients of the dynamical con-ductivity tensor σ ij ( ω ) given below. By applying a sim-ilar argument for B y , we obtain the equations for theamplitudes, (cid:18) αiωµ − σ xx ( ω ) − σ xy ( ω ) σ xy ( ω ) iωǫα − σ xx ( ω ) (cid:19) (cid:18) E TE x E TM y (cid:19) = 0 . (3)Here, µ ( ǫ ) is the permeability (permittivity) of freespace, α = r k y − ε ω c , (4)and ε ≡ ǫ/ǫ is the relative permittivity of the sur-rounding material. We assume that ε is a frequency-independent constant throughout this paper. A detailedderivation of Eqs. (3) and (4) is given in Appendix. TM TE
FIG. 1: (color online) The electromagnetic fields ( E and B )of the transverse magnetic (TM) and transverse electric (TE)modes are shown for z > z < B x ( E z ) and B y at z = 0 are relevantto the TM and TE modes, respectively. In contrast to E z , B z cannot be discontinuous at z = 0 because of the absence of amagnetic monopole. A 2DEG layer at z = 0 is expressed bya transparent sheet. We adopt the Drude model to calculate σ ij ( ω ). Themodel describes the motion of an electron acceleratedby the electric fields in an applied static magnetic field, B a . This motion is governed by the classical equation ofmotion: m ( d v /dt + v /τ ) = − e ( E + v × B a ), where m is the effective mass of the electron, τ is the relaxationtime, and v is the velocity. The solution of the equationgives, with the definition of the current j ≡ − en v ( n is the carrier density), the conductivities of the Drudemodel as σ xx ( ω ) = (1 − iωτ ) σ (1 − iωτ ) + ( ω c τ ) ,σ xy ( ω ) = − ( ω c τ ) σ (1 − iωτ ) + ( ω c τ ) , (5)where σ = ne τ /m is the static conductivity and ω c = eB az /m is the cyclotron frequency. The frequency andeigenvector of the surface plasmons are determined fromEq. (3) with Eq. (5).It has been shown by Fal’ko and Khmel’nitskii thatplasmons whose frequencies have no real part exist when ω c = 0. The off-diagonal terms of Eq. (3) disappear, sothat the TM and TE modes are decoupled completely.The TM mode satisfies the quadratic equation with re-spect to ω , 2 iωǫα − σ − iωτ = 0 . (6)In particular, when τ ≪ ǫ/ σ α , we obtain two rootscorresponding to a long-lived mode with ω ≃ − iσ | k y | / ǫ and a short-lived mode with ω ≃ − i/τ . The frequenciesof these modes have a zero real part and are expressedas ω = − iδ with a positive real number δ . The timeevolution exhibits an exponential decay (overdamped os-cillation), e − iωt = e − δt , in other words, they are non-oscillating and purely relaxational states. For the short-lived mode, the lifetime is identical to the relaxation timeof the electron, suggesting that the mode is controlledby the electron’s motion. The lifetime of the long-livedmode is inversely proportional to τ since σ is propor-tional to τ and is enhanced in the long-wavelength limit | k y | → These purely relaxational states are distinctfrom the mode extensively discussed in the literature thatappears in the collisionless limit τ ≫ ǫ/ σ α . Themode oscillates with the frequency, ω ≃ ± ω p − i τ , (7)where ω p ≡ r σ α ǫτ . (8)The positive frequency mode represents a propagatingwave with a positive (negative) velocity in the directionof y for k y > k y < αiωµ − σ − iωτ = 0 . (9)This equation has a unique solution exhibiting an over-damped oscillation, ω = − iτ + σ µ | k y | . (10)The lifetime of the TE mode is enhanced in the long-wavelength limit | k y | →
0. On the other hand, the equa-tion does not admit an underdamped oscillation with anon-zero real part of the frequency. -30 -20 -10 10 20 30-1.0-0.8-0.6-0.4-0.2
Bulk MPhigh |B az |high |B az | FIG. 2: (color online) The existence of three branches of mag-netoplasmons is shown by flows with circles in the complex ω -plane. The axes are expressed in units of ps − . The ar-rows denote the directions of the change in frequency as themagnetic field or ω c increases from 10 to 3 × s − . Weassumed that τ = 1 ps, α = 1 µ m − , σ = 10 − Ω − , and ε = 2 .
4. The lifetime of the (bulk) magnetoplasmons in thehigh magnetic field limit corresponds to the value of τ . The system supports three eigenmodes in the presenceof an external magnetic field. This is because we ob-tain the cubic equation with respect to ω , by making thedeterminant of the matrix of Eq. (3) equal to zero, as4 ǫµ (cid:8) (1 − iωτ ) + ( ω c τ ) (cid:9) − (1 − iωτ ) σ (cid:18) iωǫα + 2 αiωµ (cid:19) + σ = 0 . (11)The ω values of three solutions, which are calculated nu-merically, are shown in the complex ω -plane in Fig. 2 forthe typical case of parameters. We find that the systemsupports a purely relaxational state in the presence of anexternal magnetic field. The frequency is found analyti-cally when τ is sufficiently large ( | ω c | τ ≫
1) as ω ≃ − iτ + ω c ω p τ + µ σ α . (12)The lifetime of this purely relaxational state is elongatedby increasing | ω c | or in the long-wavelength limit. Theeigenvector is dominated by the TM component when σ /ǫ c ≪ ( | ω c | τ ) / is satisfied. The dominance of theTM component is slightly peculiar because Eq. (12) re-produces the frequency of the TE mode Eq. (10) in the limit ω c = 0. The purely relaxational state is distinctfrom the bulk magnetoplasmons with respect to the po-sitions in the frequency domain and eigenvectors. Thedispersion relation of the bulk magnetoplasmons is ob-tained by making the component that is proportional to ǫτ /µ in Eq. (11) equal to zero as ω mp ( ω c ) = ± q ω c + ω p − iτ . (13)The negative frequency mode is an unphysical mode thatshould be omitted. The eigenvectors of the modes are ahybrid of the TM and TE modes. Generally, they arecategorized as an elliptical polarization. Practically, theyare linear polarization because the TE component is dom-inant in a strong magnetic field ( σ xy ≫ σ xx ).It can be shown that the non-oscillating state foundabove starts oscillating when the state is localized. Weconsider localized electric fields of the form, E x = E x e i ( k y y − ωt ) e − βx e − α | z | ,E y = E y e i ( k y y − ωt ) e − βx e − α | z | ,E z = E z e i ( k y y − ωt ) e − βx e − α | z | . (14)Here β − is the lateral localization length from an edgeand, if β = 0, the electric fields reproduce Eq. (1). Forthe moment, we consider a positive β by limiting ourattention to the bulk of the right half-plane of x > × αiωµ (cid:16) β α (cid:17) − σ xx ( ω ) − k y ωµ βα − σ xy ( ω ) − k y ωµ βα + σ xy ( ω ) iωǫα − σ xx ( ω ) − βiωµ βα ! . (15)Each element of this matrix contains a term that is pro-portional to ω − , which is enhanced for the solution ofEq. (12) because the ω is very close to the origin of thecomplex ω -plane. This suggests that the eigenvectoris modified accordingly so that the term is suppressed.From Eq. (15) the condition on the eigenvector is readoff as βE y + ik y E x iω = 0 , (16)which is equivalent to an internal magnetic field B z be-ing suppressed in the bulk (see Eq. (A3)). By usingEq. (16), we have the equations for the amplitudes asfollows: αiωµ (cid:16) β α (cid:17) − σ xx ( ω ) − k y ωµ βα − σ xy ( ω ) σ xy ( ω ) iωǫα − σ xx ( ω ) ! (cid:18) E x E y (cid:19) = 0 . (17)The vanishing determinant of the 2 × ω : ω c + ω p − ω − iωµ σ α (1 + β α ) τ + σ µ τ α ω p − ω β α − ω p iωτ − ω c ω ω p βk y α + β = 0 . (18)There is a solution that is approximated as ω ≃ ω p ω c βα + β k y − iτ + ω c ω p τ + µ σ α (1+ β α ) . (19)The frequency has a non-zero real part, which is linear in k y . The fact that Re( ω ) /k y ∝ /ω c shows that the modeis chiral (propagation direction is dependent on the signof B az ). Note also that the group velocity is suppressedwhen | ω c | is increased and the localization length β − isfixed. All these properties of this solution are consistentwith those of the edge magnetoplasmons and Eq. (19)can reproduce Eq. (12) in the β = 0 limit. Thus, weidentify the mode with the edge magnetoplasmons. Itis worth noting that the intrinsic decay time of an edgemagnetoplasmon is derived from Eq. (19) as τ emp ≃ ω c + ω p ω p τ, (20)where µ σ / α (1 + β α ) is omitted. The ratio of τ emp to τ increases as | B az | increases, and it is expressed interms of the angular frequencies of the magnetoplasmonsas τ emp /τ = ω mp ( ω c ) /ω mp (0). Note that τ can dependon | B az | , and the dependence can be determined fromthe lifetime of the magnetoplasmons (see Eq. (13)). Equation (20) can be used to explain a recent experi-ment on the edge magnetoplasmons in graphene reportedby Yan et al. . In Fig. 3, magnetic field dependences ofthe full width at half maximum (FWHM) of bulk andedge magnetoplasmons are indicated by errorbars withfilled and empty squares, respectively. These are the ex-perimental data taken from Fig. 2(D) in Ref. 6. Theplot of 2 /cτ emp gives errorbars with circles, where τ inEq. (20) is taken from the experiment (the errorbars withfilled squares). The close agreement between the posi-tions of circles and empty squares supports the validity ofour result. Moreover, τ emp is elongated by decreasing | k y | through ω p , which is consistent with a recent experimenton the dissipation mechanism in graphene. Our derivation of the intrinsic lifetime of the edge mag-netoplasmons does not assume the details of the bound-ary of a 2DEG and therefore has a wide application. Thisis in contrast to the analyses of Volkov and Mikhailov,in which a sharp electron density profile at the bound-ary (sharp edge) is assumed. The edges of graphene andInAs meet this assumption, while GaAs might not be-cause of the existence of a depletion layer several mi-crometers thick. Acoustic types of edge magnetoplas-mons have been predicted for such smooth edge. Note
Edge MP exp.Bulk MP exp.Edge MP theoryB az field (Tesla) F W H M ( c m - ) FIG. 3: (color online) Equation (20) is applied to a recentexperiment on the edge magnetoplasmons in graphene re-ported by Yan et al. The close agreement between the cir-cle and empty square plots supports the validity of Eq. (20).The dashed curves are the plots of 69 p ( ω c + ω p ) /ω p cm − and 69 p ω p / ( ω c + ω p ) cm − . The errorbars with circles areshifted slightly horizontally from the proper values of the mag-netic field in order to avoid overlap between the plots. that since plasmon is the hybrid of electrons and elec-tromagnetic fields, it is difficult to identify the lifetimeof edge magnetoplasmon with the lifetime of electrons atonly the edge channel. Our formulation does not assumethe specific properties of the electric edge states, such asthe absence of back scattering, but it yields close agree-ment between Eq. (20) and the result in Ref. 6. Thisfact suggests that because the experiment by Yan et al. is performed in the classical Hall effect region, many elec-tronic states including not only the edge states but also(bulk) states near the edge (up to several micrometersfrom the edge) are participating in the dynamics of theedge magnetoplasmon. The present model suggests anintriguing physical interpretation of the longer lifetimebased on the hybridization of the TE and TM modes. Adeviation from Eq. (20) may appear in the quantum Halleffect region and it can be attributed to the contributionof the specific properties of the electric edge states.The finite real part of the frequency of an edge mag-netoplasmon originates from the mixing of E x and E y .Namely, when we make a purely relaxational state lo-calized by a non-zero β , the eigenvector of the state ismodified according to βE y + ik y E x = 0 for x >
0. Animportant feature of the modified eigenvector for x < βE y + ik y E x = 0 does not need to be satis-fied, can be grasped pictorially without a mathematicalcalculation. Suppose that as shown in Fig. 4(a), a purelyrelaxational state, which is a TM dominant mode, existsfor a finite period of time in a periodic system withoutan edge. Note that the magnetic field B x at z = 0+ ispointing in the opposite direction to that at z = 0 − . If weintroduce the edge along the y -axis (by cutting the layer)in Fig. 4(b), the magnetic fields at z > z < ∇ · B = 0. Thus, the magnetic field TMTE TE TM (a)(b)(c)
FIG. 4: (color online) (a) The magnetic field of a purely re-laxational TM dominant state with β = 0 is shown for z > z <
0. The sheet current j y δ ( z ) that causes the disconti-nuity of B x , is represented by a transparent layer. (b) Whenwe introduce the edge along the y -axis, the magnetic fields at z > z < z -component, which is a locally inducedTE mode. (c) The edge-induced hybridization of the TE andTM modes results in an edge magnetoplasmon with a chiralpropagation property. must have a non-zero z -component at the edge, which isa locally induced TE mode. We may regard the edgemagnetoplasmon as a composite of the TE mode at theedge and the spatially decaying TM mode in the bulk, asshown in Fig. 4(c).Since a TE mode is inevitably hybridized with thepurely relaxational state when the state changes intoan edge magnetoplasmon, the fact that a TE mode ex-hibits anomalous behavior in graphene is noteworthy.Mikhailov and Ziegler pointed out that the imaginarypart of the dynamical conductivity of graphene can benegative for a specific frequency, because an inter-band transition contributes to the dynamical conductiv-ity, while the Drude model only accounts for an intrabandtransition. As a result, they predict that graphene cansupport a TE mode for a special frequency (even with-out an external magnetic field). We can easily see fromEq. (9) that an oscillating TE mode can appear when theimaginary part of the dynamical conductivity is a neg-ative number. Bordag and Pirozhenko argued that theexistence of an infinitesimal mass gap in graphene leadsto a special TE mode that propagates at the speed oflight. In summary, the peculiarities of a purely relaxational state in the bulk are partly eliminated by knowing theirrelationship to edge magnetoplasmons, which have con-stituted the theme of various published reports.
Thestrange behavior found for the state, such as the intrin-sic long lifetime being proportional to the square of theapplied magnetic field, is our original conclusion that hasnot been taken into account before. Our result wherebythe eigenvector is TM dominant with a very long-lived E z component has an advantage in that it detects the signalvia an electrode in a transient manner. The local mix-ing with the TE mode may also imply that ferromagneticelectrodes can excite the edge magnetoplasmons.
Acknowledgments
K. S. is indebted to N. Kumada and H. Sumikura fordiscussions.
Appendix A: Derivation of Eq. (17)
By applying Faraday’s law to the electric fieldsEq. (14), we obtain the magnetic fields for z > iωB x = ik y E z + αE y , (A1) iωB y = − αE x + βE z , (A2) iωB z = − βE y − ik y E x . (A3)For z <
0, the magnetic fields are given by replacing α with − α and E z with − E z on the right-hand side. As aresult, B x and B y change their signs at z = 0. Note thatthe sign change of E z imposed at z = 0 is still consistentwith Gauss’s law, which gives − βE x + ik y E y − αE z = 0 . (A4)It is useful to write Eqs. (A1), (A2), and (A3) in the formof a 3 × iω B = M E where E = t ( E x , E y , E z ), B = t ( B x , B y , B z ), and M = α ik y − α β − ik y − β . (A5)By applying Amp´ere’s circuital law for free space ( z = 0)where the charged current is absent ( j = 0), c ∇ × B = ε∂ E /∂t , we obtain (cid:18) − iωεc (cid:19) E = M B . (A6)By multiplying iω with both sides and using iω B = M E ,we have [ ε ω c − M ] E = 0. Thus, a non-vanishing electricfield is possible when det[ ε ω c − M ] = 0, namely, when εω (cid:18) − k y + α + β + εω c (cid:19) = 0 (A7)is satisfied.The boundary conditions for the magnetic fields B x and B y at z = 0 are expressed by c ( B y ( z = 0 − ) − B y ( z = 0 + )) = j x ( x, y ) ǫ , (A8) c ( B x ( z = 0 + ) − B x ( z = 0 − )) = j y ( x, y ) ǫ . (A9) Putting Eqs. (A1) and (A2) into these boundary condi-tions and using Eqs. (A3), (A4), and (A7), we obtainEq. (15) or αiωµ (cid:16) β α (cid:17) − σ xx ( ω ) − k y ωµ βα − σ xy ( ω ) − k y ωµ βα + σ xy ( ω ) iωǫα − σ xx ( ω ) − βiωµ βα ! (cid:18) E x E y (cid:19) = 0 . (A10)This equation reproduces Eq. (17) when βE y + ik y E x iω = 0 . (A11)It is noted that the determinant of the matrix ofEq. (A10) is independent of the variable β . Indeed, bysetting β = − ik x in Eq. (14), it is easily understoodthat the inclusion of β merely changes the propagationdirection. Thus, the solutions are given by Eqs. (12)and (13), and therefore Eq. (A10) fails to reproduces theedge magnetoplasmons. This clarifies the importance ofthe condition Eq. (A11).It is possible to estimate the value of β using Eq. (A11).By putting the eigenstate of Eq. (17) on Eq. (A11), weobtain β ≃ k y ω c (cid:18) ω c ω p (cid:19) ω + i k y ω c τ , (A12) which is correct up to the first order of ω . By substitutingEq. (19) for ω , we obtain α + β ≃ ω c + ω p ω c k y . (A13)Thus, β is determined as a function of α . We note thatthe α value is determined from ω p or ω mp ( ω c ). It is alsoworth noting that with Eq. (A12), we can obtain ω asa function of α by eliminating β from Eq. (A7). Thecalculated ω reproduces Eq. (19). ∗ Electronic address: [email protected] S. J. Allen, H. L. St¨ormer, and J. C. M. Hwang, PhysicalReview B, , 4875 (1983), ISSN 0163-1829. D. C. Glattli, E. Y. Andrei, G. Deville, J. Poitrenaud,and F. I. B. Williams, Physical Review Letters, , 1710(1985), ISSN 0031-9007. I. Grodnensky, D. Heitmann, and K. von Klitzing, Physi-cal Review Letters, , 1019 (1991), ISSN 0031-9007. R. C. Ashoori, H. L. Stormer, L. N. Pfeiffer, K. W. Bald-win, and K. West, Physical Review B, , 3894 (1992),ISSN 0163-1829. M. Tonouchi, T. Miyasato, P. Hawker, T. Cheng, andV. Rampton, Journal of the Physical Society of Japan, ,4499 (1994), ISSN 0031-9015. H. Yan, Z. Li, X. Li, W. Zhu, P. Avouris, and F. Xia,Nano letters, , 3766 (2012), ISSN 1530-6992. I. Petkovi´c, F. I. B. Williams, K. Bennaceur, F. Portier,P. Roche, and D. C. Glattli, Physical Review Letters, ,016801 (2013), ISSN 0031-9007. N. Kumada, S. Tanabe, H. Hibino, H. Kamata,M. Hashisaka, K. Muraki, and T. Fujisawa, Nature com-munications, , 1363 (2013), ISSN 2041-1723. V. A. Volkov and S. A. Mikhailov, Sov. Phys. JETP, ,1639 (1988). B. Noble,
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300 is the Fermi velocity of graphene).It is noted that by using the reported values (Fermi en-ergy | E F | = 0 . ε = 2 . ω p ≃
130 cm − by setting thewavevector | k | = 0 . µm − . An extension of the work by Volkov and Mikhailov gives amatrix Wiener-Hopf equation, which is not solved in gen-eral. However, by using Eq. (16) we can find that the mag-netic field x < The purely relaxational state can exist in a gated sam-ple. When a metal gate is placed on the dielectric me-dia at a distance d from the layer, the corresponding fre-quency can be calculated by replacing ǫ and µ in Eq. (12)with ǫ (1 + coth( αd )) / µ / (1 + α m + α tanh( αd ) α + α m tanh( αd ) ), respec-tively, where α m is the inverse of the localization length ofa metal, which may be taken to be ∞∞