October, 2001 Quantum dots and wells, mesoscopic networks 65 rule out this possibility. In early reports on CDW dynamics, Mesoscopic physics on graphs NDR has also been observed in NbSe3 crystals at low temperatures (45 K) accompanied by unusually large 1/f noise [17]. At even lower temperatures, these samples show G Montambaux switching events near threshold. The measurements [17] are Abstract. This report is a summary of recent work on the performed with voltage probes spaced at macroscopic dis- properties of phase coherent diffusive conductors, especially in tances (L 4 100 mm). For L > 5 mm, we have never seen NDR the geometry of networks Ð also called graphs Ð made of nor switching events in our TaS3 crystals in the temperature quasi-1D diffusive wires. These properties are written as a range studied (T > 90 K). function of the spectral determinant of the diffusion equation We believe that the N(D)R we observe, originates from (the product of its eigenvalues). For a network with N nodes, the response to local CDW deformations. Assuming that this spectral determinant is related to the determinant of an there are a few strong pinning centers or line dislocations in N  N matrix which describes the connectivity of the network. our crystal, strong deformations in the strain profile may be I also consider the transmission through networks made of 1D present around these regions. Strain leads to a shift of the ballistic wires and show how the transmission coefficient can be chemical potential and this shift can be either up or down written in terms of an N  N matrix very similar to the above depending on the direction of sliding. For a semiconducting one. Finally I present a few considerations on the relation CDW with the Fermi level in the middle of the gap, a shift of between the magnetism of noninteracting systems and the the chemical potential leads to an increase of quasi-particles magnetism of interacting diffusive systems. (either electrons or holes) and consequently to a decrease of the quasi-particle resistance. Such a resistance decrease may 1. Return probability then produce regions with NDR as argued previously by Latyshev et al. [18] to explain NDR in their data on partly Transport and thermodynamic properties of phase coherent irradiated macroscopic samples. However, this reasoning disordered conductors can be described in a simple unified cannot be used to explain a negative resistance: the decrease way: all can be related to the classical return probability P t† of the quasi-particle resistance cannot change the sign of the for a diffusive particle. I consider diffusive conductors, for resistance. Most likely, nonequilibrium processes between the which the mean free path le is much larger than the distance CDW and the quasi-particles must also play a role in this new between electrons: kFle 4 1. The return probability has two phenomenon as well. No theory is available as yet. components, a purely classical one (diffuson) and an inter- Acknowledgements. The fabrication of the wire structure ference term which results from interferences between pairs of (Fig. 1b) has been performed in close collaboration with time-reversed trajectories (Cooperon). The classical term is Yu I Latyshev, B Pannetier and P Monceau. The o-TaS3 solution of the differential equation crystals were provided by R Thorne. The authors are grateful 0 0 to Yu V Nazarov, Yu I Latyshev, S V Zaitsev-Zotov and ‰io DDŠPcl r; r ; o†ˆd rr†; 1† S N Artemenko for useful discussions. This work is supported by the Netherlands Foundation for Fundamental Research and the interference term is solution of the equation [1]: on Matter (FOM). HSJvdZ is supported by the Dutch Royal "# Academy of Arts and Sciences (KNAW). 2ieA 2 g io D H ‡ P r; r0; o†ˆd rr0†; 2† hc int References 0 1. GruÈ ner G Density Waves in Solids (Reading, Mass.: Addison- whose solution has to be taken at r ˆ r. D is the diffusion 2 Wesley, 1994) coefficient. The scattering rate g ˆ 1=tf ˆ D=Lf describes 2. Peierls R, in Quantum Theory of Solids (Oxford: Oxford Univ. Press, the breaking of phase coherence. Lf is the phase coherence 1955) p. 108 length and tf is the phase coherence time. Finally, the space 3. Mantel O C et al. Synthetic Metals 103 2612 (1999) integrated (dimensionless) return probability is defined as 4. Gor'kov L P Pis'ma Zh. Eksp. Teor. Fiz. 38 76 (1983) [JETP Lett. 38 87 (1983)] 5. Ong N P, Maki K Phys. Rev. B 32 6582 (1985) P t†ˆ P r;r;t†dr: 6. Itkis M E, Emerling B M, Brill J W Phys. Rev. B 52 R11545 (1995) 7. DiCarlo D et al. Phys. Rev. Lett. 70 845 (1993) 8. Requardt H et al. Phys. Rev. Lett. 80 5631 (1998) The weak-localization correction to the conductance can be 9. Gill J C Physica B 143 49 (1986) written as [1] 10. Borodin D V,Zaitsev-Zotov S V, Nad' F Ya Zh. Eksp. Teor. Fiz. 93  1394 (1987) [Sov. Phys. JETP 66 793 (1987)] e2D 1 t Ds ˆs P t† exp gt†exp dt; 3† 11. Gill J C Phys. Rev. Lett. 70 331 (1993) phO int t 12. Maher M P et al. Phys. Rev. B 52 13850 (1995) 0 e 13. Lemay S G et al. Phys. Rev. B 57 12781 (1998) 14. McCarten J et al. Phys. Rev. B 46 4456 (1992) Ois the volume and s is the spin degeneracy. The contribution 15. Ramakrishna S et al. Phys. Rev. Lett. 68 2066 (1992); Ramakrishna of the return probability is integrated between te, the smallest S Phys. Rev. B 48 5025 (1993) time for diffusion, and the phase coherence time tf ˆ 1=g. 16. Mantel O C et al. Phys. Rev. Lett. 84 538 (2000) Similarly, the variance of the conductance fluctuations at 17. See, e.g., Hall R P, Sherwin M, Zettl A Phys. Rev. Lett. 52 2293 T ˆ 0 K is given by (1984) 18. Latyshev Yu I, Pogotovsky N P, Artemenko S N Synthetic Metals 29 F415 (1989) G Montambaux Laboratoire de Physique des Solides, associe au CNRS, Universite Paris-Sud, 91405 Orsay Cedex, France 66 Chernogolovka 2000: Mesoscopic and strongly correlated electron systems Usp. Fiz. Nauk (Suppl.) 171 (10)

determinant, so that the physical quantities described above  2 3 e2D read [8]: hds2i ˆ s b phO 2 1    e D q t Ds ˆ s ln Sd g† ; 10†  ‰Pcl t† ‡ Pint t†Š exp gt† exp t dt ; phO qg 0 te   3 e2D 2 q2 4† hds2i ˆ s ln S g† ; 11† b phO qg2 d b ˆ 1 in the absence of a magnetic field; b ˆ 2 if the field completely breaks the time reversal symmetry. The interac- h2 ‡1 q2 M2 ˆ dg g g † ln S g † B ; 12† typ 2 1 1 2 d 1 0 tion contribution to the average magnetization of a disor- 2p g qB dered electron gas can also be written as a function of field- ‡1 dependent part of the return probability. For a screened l0h q hMeei ˆ dg1 ln Sd g1† : 13† interaction U r r0† ˆ Ud r r0†, one has [2, 3] p g qB

l0h q These expressions are quite general, strictly equivalent to hMeei ˆ p qB    expressions (3) ± (6). On a graph made of quasi-1D diffusive 1 t dt  P t; B† exp gt† exp ; 5† wires, the spectral determinant can be calculated explicitly. By int 2 0 te t solving the diffusion equation on each link, and then imposing Kirchhoff-type conditions at the nodes of the where l0 ˆ Ur0 is the interaction parameter. Considering graph, the problem can be reduced to the solution of a higher corrections in the Cooper channel leads to a ladder system of N linear equations relating the eigenvalues at the summation [4, 5], so that l0 should be replaced by N nodes. Let us introduce the N  N matrix M [9]: l t† ˆ l0=‰1 ‡ l0 ln EFt†Š ˆ 1= ln T0t† where T0 is defined as X   T ˆ E exp 1=l †. lab exp iyab† 0 F 0 Maa ˆ coth ; Mab ˆ Á : 14† 2 L sinh l =L Finally the typical magnetization Mtyp, defined as Mtyp ˆ b f ab f 2 2 hM i hMi , is given by (neglecting the interaction) [2, 7] P The sum extends to all the nodes b connected to the h2 b M2 ˆ node a; lab is the length of the link between a and b. The off- typ 2 2p    diagonal coefficient Mab is non zero only if „there is a link ‡1 t dt connecting the nodes a and b; y ˆ 4p=f † b A Á dl is the  ‰P00 t; B† P00 t; 0†Š exp gt† exp ; ab 0 a int cl 3 0 te t circulation of the vector potential between a and b; NB is the 6† number of links in the graph. It can then be shown that for an isolated graph the spectral determinant Sd is given by [8] 2 where P00 t; B† ˆ q P t; B†=qB2. In the geometry of a quasi-     L NBNY l 1D ring, the magnetization M is proportional to the persistent S ˆ f sinh ab det M : 15† current M ˆ IS where S is the area of the ring. Eqns (5,6) can d L L 0 ab† f thus be used to get the persistent current of a diffusive ring from the known expression of P t† is such a geometry [3]. L0 is an arbitrary length independent of Lf. We have thus transformed the spectral determinant which is an infinite 2. Diffusion on graphs product into a finite product related to det M. The properties of this spectral determinant have been studied in details in We wish to calculate the above quantities in any structure Ref. [10], in particular, the role of the boundary conditions made of quasi-1D wires. To this end, we reformulate the has been considered and a trace expansion of the spectral different results. We first note that the quantities of interest determinant has been obtained. As a physical important have all the same structure. They are proportional to result, it has been found that when isolated rings are

1 connected into an array of rings, the average magnetization taP t† exp gt† dt ; 7† per ring is reduced by a factor r given by [8] 0   Y 2 P r ˆ ; 16† zi where P t† ˆ n exp Ent† and En are the eigenvalues of the i diffusion equations (1) or (2). The time integral of P t† can be straightforwardly written in terms of a quantity called the where zi are the coordinates of the nodes belonging to each spectral determinant Sd g†: ring.

1 X 1 q P  dtP t† ˆ ˆ ln Sd g† ; 8† 3. Transmission through ballistic quantum E ‡ g qg 0 n n graphs where Sd g† is, within a multiplicative constant independent We now consider a graph made of 1D ballistic wires. There of g, are N nodes connected to Nin input channels and to Nout Y output channels. The total transmission coefficient T from Sd g† ˆ g ‡ En† : 9† the left to the right reservoirs is given by n X 2 Using standard properties of Laplace transforms, the above T ˆ jtijj ; 17† time integrals can be rewritten in terms of the spectral i;j October, 2001 Quantum dots and wells, mesoscopic networks 67

where i 2 ‰1; NinŠ denote the i th input channel and 2e ! e. Comparing Eqns (22), (23) with Eqn (13), we can now j 2 ‰N Nout ‡ 1; NŠ denote the j th output channel. The formally relate M0 and the HF magnetization hMeei of the transmission coefficient (17) is the sum of each individual same diffusive system: transmission coefficient obtained by injecting a wave packet h  i 1 EF in the i th channel. It is assumed that there is no phase M0 ˆ~ lim Im hMeei g ˆ i0 ; 24† relationship between electrons in the different channels. l0!0 l0 h Solving SchroÈ dinger equation for the wave function c on each bond of the network, and writing current conservation at where the sign ˆ~ means that the two quantities are equal, the nodes, one gets provided the substitutions D ! h= 2m† and 2e ! e are made. X This relation can be inverted to obtain Maaca ‡ Mabcb ˆ 0 ; 18† l 1 M E† b hM i ˆ~ 0 0 dE : 25† ee p E ‡ hg P 0 where b extends to the nodes b connected to the node a. Current conservation at the input node i writes Taking into account the renormalization of the interaction X parameter in the Cooper channel, Miici ‡ Mibcb ˆ i 1 rii† ; 19† b l0 1 l0 ! l E† ˆ ˆ ; 26† 1 ‡ l0 ln EF=E† ln T0=E† and for an output node j, one has X where T0 is defined as T0 ˆ EF exp 1=l0†, the energy depen- Mjjcj ‡ Mjbcb ˆ itij ; 20† dence of the parameter l E† can be incorporated exactly in the b integral so that:

1 with c ˆ 1 ‡ r and c ˆ t ; M is an N  N matrix whose 1 M0 E† i ii j ij hMeei ˆ~ l E† dE : 27† elements are p 0 E ‡ hg X exp iyab As a simple example, consider the Landau spinless suscept- Maa ˆ cot klab† ; Mab ˆ : 21† sin klab 2 b ibility in 2D, given by w0 ˆ e =24pm ˆ wL=2. From Eqn (27), one gets the phase coherent contribution of the Here k is th„e wave vector of the incident electron; electron ± electron interaction to the susceptibility [13]: y ˆ 2p=f † b A Á dl is the circulation of the vector ab 0 a E t ln T t =h† potential between a and b. The equations (18) ± (20) consti- w ' 4jw j F e ln 0 j ; 28† ee L h ln T t =h† tute a linear system of N equations from which the tij can be 0 e calculated. The total transmission coefficient T k† is finally obtained from Eqn (17) by considering all the input channels. where wL is the Landau susceptibility. An ultraviolet cutoff This formalism has been used recently to calculate the 1=te has been added to cure the divergence at large energy. transmission coefficient of regular networks, in particular of the so-called T 3 network which exhibits the Aharonov ± 5. Conclusions Bohm cage effect, i.e. the absence of transmission for half flux quantum per plaquette [11]. In conclusion, we have shown how to relate phase coherent transport and thermodynamic properties to the return 4. Landau diamagnetism and magnetization of probability for a diffusive particle. It is then possible to an interacting electron gas calculate straightforwardly these quantities by simple inte- grals of this return probability in simple geometries. For Finally, we wish to emphasize an interesting correspondence networks made of diffusive wires, we have developed a between the magnetization of a phase coherent interacting formalism which relates directly the persistent current, and diffusive system and the grand canonical magnetization M0 of the transport properties to the determinant of a matrix M the corresponding noninteracting clean system. The latter can describing the connectivity of the graph. From a loop also be written in term of a spectral determinant. The grand expansion of this determinant, simple predictions for the canonical magnetization M0 is given quite generally by persistent current in any geometry can now be compared with forthcoming experiments on connected and discon- qO q 0 M ˆ ˆ dEN E† ; 22† nected rings. Then we have shown that the transmission 0 qB qB EF coefficient of a network of one-dimensional ballistic wires can be written in terms of a connectivity matrix very similar to where the integrated DOS can be rewritten as M with the substitution kL ! iL=Lf. Finally, we have found a correspondence between the phase coherent contribution to 1 X 1 N E† ˆ Im ln E E † ˆ Im ln S E † : 23† the orbital magnetism of a disordered interacting system and p m ‡ p ‡ Em the orbital response of the corresponding clean noninteract- Q ing system. Here E ˆ E ‡ i0, S E† ˆ E E† / S g ˆ E=h†, where Acknowledgements. Parts of this work and related studies ‡ Em m d Em are the eigenvalues of the Schrodinger equation. For a have been done in collaboration with E Akkermans, clean system these eigenvalues are the same as those of the A Comtet, J Desbois, B DoucË ot, M Pascaud, C Texier, and diffusion equation, with the substitutions D ! h= 2m† and J Vidal. 68 Chernogolovka 2000: Mesoscopic and strongly correlated electron systems Usp. Fiz. Nauk (Suppl.) 171 (10)

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