Presence of asymmetric noise in multi-terminal chaotic cavities
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A ug Presence of asymmetric noise in multi-terminal chaotic cavities
A. L. R. Barbosa , J. G. G. S. Ramos , and D. Bazeia Unidade Acadˆemica de Ensino a Distˆacia e Tecnologia,Universidade Federal Rural de Pernambuco, 52171-900 Recife, PE, Brazil Departamento de F´ısica, Universidade Federal dea Para´ıba, 58051-900 Jo˜ao Pessoa, PB, Brazil (Dated: September 15, 2018)This work deals with chaotic quantum dot connected to two and four leads. We use standard dia-grammatic procedure to integrate on the unitary group, to study the main term in the semiclassicalexpansion of the noise in the three pure Wigner-Dyson ensembles at finite frequency and temper-ature, in the noninteracting and interacting regimes. We investigate several limits, related to thetemperature and the potential difference in the leads, in the presence of a capacitive environment.At the thermal crossover regime, we obtain general expressions described in terms of parametersthat can be controlled experimentally. As an interesting result, we show the appearance of asym-metries in the noise, controlled by the topology of the cavity and the number of open channels inthe corresponding terminals.
PACS numbers: 73.23.-b, 73.21.La, 05.45.Mt
I. INTRODUCTION
The study of coherent charge transport throughchaotic mesoscopic quantum dots is of current interestin physics [1–5]. Subtle effects appear from the inter-ference originated through the multiple coherent scatter-ing [6, 7], which strongly depend on external parame-ters such as the magnetic field, the spin-orbit coupling,the topology, the temperature and other phenomenologi-cal parameters that can be controlled experimentally [8].An important observable of the quantum transport whichstrongly depends on the finite size of the chaotic quantumdot is the shot noise power, the second cumulant of thecounting statistics of the electrons at zero temperature.Experimental measurements of the shot noise power andof other cumulants have been performed in [9, 10]. Theshot noise provides direct information about the discrete-ness of matter [11–13], so it is different from the thermalnoise [1, 14]. In mesoscopic systems, the two source ofnoises contribute for the quantum transport of electronsthrough the cavity at finite temperature.Nontrivial properties of the noise peaks indicate a sub-tle combination of both spatial and temporal coherencein the electronic propagation through the system. Inchaotic quantum dots, in accordance with the results ofRef. [15], asymmetries in the spectral density of the noisein the Kondo regime are obtained in terms of the tensionin the electron reservoirs. In this regime, the asymmetrywas shown to strongly depend on the AC frequency whichirradiate the chaotic quantum dots connected to two ter-minals. The presence of the AC frequency also inducesasymmetric peaks in the derivative of the noise. Anothereffect that shows radical change in the shot noise powerof a chaotic quantum dot was pointed out in [16, 17].In the Landauer-Buttiker framework, a particularlyinteresting characteristic of the quantum transport inchaotic mesoscopic systems is the dependence with thegeometry. A critical example which shows the impor-tance of the geometry of the chaotic mesoscopic sys- tems appeared in Ref. [18], where the authors identifieda change in the correction of the weak localization of theconductance in the case of a multi-terminal network madeof quasi-one-dimensional diffusive wires. The effect thereidentified is directly related to the geometry: for a sys-tem of N wires connected to the center of the conductingquasi-one-dimensional region, the Cooperon correction tothe quantum interference reads δT = 1 / N/ − δT vanishes for N = 4, a change of signal is induced geo-metrically, indicating a transition depletion-enhancementfor the conductance due to the quantum interference.Other recent investigations concerning topology ofquantum dots connected to several terminals appearedin [19]. A key issue concerned the time-dependent fluc-tuations in the electronic current injected from reservoirswith non-equilibrium spin accumulation into mesoscopicconductors. There, using topology the authors could sug-gest an interesting way to measure spin accumulation inthe regime of vanishing frequency.The above results motivated us to study issues relatedto the topology of the mesoscopic system, concerningthe number of leads and open channels connected to thechaotic quantum dot. More specifically, in the currentwork we show the appearance of an asymmetry in themain term in the semiclassical expansion of the noisepower, as a function of the potential in each reservoirand the number of open channels in the leads, in termsof parameters that can be controlled experimentally.We study the chaotic quantum dot depicted in Fig. 1,searching for effects due to the finite temperature, thedifference of potential among the leads and the influenceof the frequency of an AC current on the noise. In themesoscopic system, each lead is connected to an elec-tronic reservoir with µ i = eV i , i = 1 , , , and 4, at tem-perature T . Here we take V = V − V and V = V − V .Thermal effects appear through the presence of tempera-ture in the electron reservoirs, and lead us to interestingscenarios, in which the chaotic cavity gives rise to thecrossover between the thermal noise, present in electrical FIG. 1: Illustration of the chaotic quantum dot connected tofour leads, with C standing for the capacitive environment.We use V = V − V and V = V − V to identify thedifference of potentials between the leads 1 and 2, and 3 and4, respectively. circuits at high temperatures, and the shot noise power,induced by the quantum effects that appear in electroniccircuits at very low temperatures [1, 8, 17, 20]. Also, oneuses C to represent the capacitive environment, whichis introduced to generate interaction among the chargecarriers inside the quantum dot [1, 7].The investigation starts using diagrammatic approachto obtain general expressions for the noise power in themulti-terminal chaotic cavity. We use the results to de-velop specific expressions and, at the very end of thestudy, we comment on several plotted results that showa surprising phenomenology and a rich manner to mea-sure the noise power. For pedagogical reasons, we orga-nize the work as follows: In the next Sec. II, we intro-duce the current-current correlation function, using thescattering theory for quantum transport and the noise interms of the scattering matrix, to describe charge trans-port through the chaotic quantum dot. We use the dia-grammatic procedure to calculate averages over the uni-tary group [21, 22]. The investigation follows in Sec. III,where we consider the noninteracting case, with vanish-ing frequency. We consider the interacting case in Sec.IV, where we show explicitly that the equations there ob-tained for the noise satisfy the current conservation lawif one substitutes the dwell time, τ D , by the charge re-laxation time. Finally, we end the investigation in Sec.V, where we deal with the rich phenomenology of thenoise in the multi-terminal case, including the surprisingeffects of the asymmetry which appears controlled by theextra terminals. The main results of the current work areshown in Figs. 2, 3, 4, 5, 6, 7, and 8, and we end thepaper in Sec. VI, where we include our comments andconclusions. II. SCATTERING THEORY FOR QUANTUMTRANSPORT
We start considering the time-dependent current ˆ I γ ( t )at lead γ , for γ = 1 , , ..., m , with m being the num-ber of leads connected to the chaotic quantum dots.Within the framework of the scattering theory for quan-tum transport, the current-current correlation functioncan be written in the form [1] h δ ˆ I γ ( t ) δ ˆ I α (0) i = Z dw π e − iwt S γα ( w ) , (1)where δ ˆ I α ( t ) ≡ ˆ I α ( t ) + h ˆ I α ( t ) i and the noise in the ab-sence of interaction is S γα ( w ) = X ν,ρ e h Z dε Tr { A ργ ( ε, ε + ¯ hw ) × A να ( ε + ¯ hw, ε ) }× { f ν ( ε ) [1 − f ρ ( ε + ¯ hw )]+ f ρ ( ε ) [1 − f ν ( ε + ¯ hw )] } . (2)The matrix A να ( ε, ε ′ ) is the current matrix, and it is de-fined as A να ( ε, ε ′ ) = 1 α ν − ν S † ( ε )1 α S ( ε ′ ), where S ( ε )is the scattering matrix, which depends on the energy ε and describes the charge transport through the circuit.Also, 1 α is the projection matrix, which projects onto thelead α , and f α ( ε ) = (1 + exp [ β ( ε − µ α )]) − representsthe Fermi distribution function, related to the thermalreservoir connected to the lead α .The scattering matrix S ( ε ) used to describe the meso-scopic system is uniformly distributed over the ortogonalgroup, if time reversal and spin rotation symmetries areboth present in the system, or over the unitary group,if the time reversal symmetry is broken by the actionof a strong external magnetic field, or yet over the sim-pletic group, if the spin rotation symmetry is broken bythe action of intense spin-orbit interaction [3]. To obtainthe average of the noise, according to Eq. (2), we haveto calculate the average of the product of two and fourscattering matrices S ( ε ). Since we are interested in themain term for the semiclassical expansion of the noise,each ensemble gives the same average value, discardinghigher order contributions. As an efficient way to get tosuch average value, we use the diagrammatic proceduredeveloped in Ref. [21].This diagrammatic procedure requires that we appro-priately parametrize the scattering matrix. We follow[22, 23], and we parametrize S ( ε ) within the stub model,in a manner which allows including external fields andother relevant phenomenological parameters: S ( ε ) = P [1 − U Qr ( ε ) Q T ] − U P T . (3)The matrix U is distributed in one of the Wigner-Dysonensembles, of dimension M × M , where M = P mγ N γ . P and Q are projection matrices, ( M + N s ) × M and M × N s , respectively. Also, r ( ε ) = e iε Φ /M is the matrixwhich describes the stub, coupled to the chaotic quan-tum dot of dimension N s × N s , with N s standing for thenumber of open channels in the stub. The matrix Φ isHermitian, positive, such that φ = TrΦ. In the reference[24], the authors show that φ is a parameter related tothe average density of modes. Thus, we can associate thestub to specific time scale, to characterize the lifetime ofthe metastable electronic modes in the chaotic quantumdot, that is, we can write φ = τ D M/ ¯ h , where τ D is thedwell time.In the limit M ≫
1, we can expand S ( ε ) in power of U .Performing the diagrammatic procedure, we can obtainthe average of the traces of the two and four scatteringmatrices and it is possible to verify that only the lad- der diagrams (known as difusons) contribute to the mainsemiclassical term. The diagrams for the average of thetrace of product of two scattering matrices S ( ε ) can befound in [21], while the diagrams for the average of thetrace of product of four scattering matrices are found in[22]. With this, we can obtain for a chaotic quantum dotconnected to several leads the following results: h Tr [ A να ( ε, ε ′ )] i = δ αν N α − N α N ν M h − i ( ε ′ − ε ) τ D ¯ h i , (4)and h Tr (cid:2) ρ S † ( ε )1 γ S ( ε ′ )1 ν S † ( ε ′′ )1 α S ( ε ′′′ ) (cid:3) i = N γ N ρ M N α h − i ( ε ′ − ε ) τ D ¯ h i h − i ( ε ′′′ − ε ′′ ) τ D ¯ h i δ ρν + N ν h − i ( ε ′′′ − ε ) τ D ¯ h i h − i ( ε ′ − ε ′′ ) τ D ¯ h i δ γα + N α N ν M h − i ( ε ′′′ + ε ′ − ε ′′ − ε ) τ D ¯ h ih − i ( ε ′ − ε ) τ D ¯ h ih − i ( ε ′′′ − ε ′′ ) τ D ¯ h ih − i ( ε ′′′ − ε ) τ D ¯ h ih − i ( ε ′ − ε ′′ ) τ D ¯ h i , (5)where α, γ, ν, ρ = 1 , , ...m . Expressions like (4) and (5)were obtained before in Refs. [24] and [20], respectively,for chaotic quantum dots connected to two leads. III. THE CASE OF VANISHING FREQUENCY
Let us now investigate the noise power associated to achaotic quantum dot connected to four leads, as shown in Fig. 1, in the regime of vanishing frequency. We firstsubstitute the above results (4) and (5) into Eq. (2) andthen take the limit w →
0. Since the general expressionis awkward, we focus the study on some cases of currentinterest, similar to the ones investigated in [8, 20]. Thefirst case concerns the regime where k B T, eV ≫ eV , inwhich V , the difference of potential between the leads3 and 4, can be seen as a perturbation. In this case wecan write S ik (0) = e h k B T N i ( δ ik M − N k ) M ( N + N )( N + N ) × (cid:26)(cid:20) N N ( N + N )( N + N ) (cid:21) eV k B T coth (cid:18) eV k B T (cid:19) + eV k B T csch (cid:18) eV k B T (cid:19) + (cid:20) N + N ) + ( N + N ) − N N ( N + N )( N + N ) (cid:21)(cid:27) , (6)where i, k = { , , , } and M = P i =1 , N i . The aboveresult (6) is in accordance with charge conservation: S kk (0) = − P i =1 ,i = k S ki (0) = − P i =1 ,i = k S ik (0). Wecan also verify that, if one sets N = N = 0, that is, ifone closes the channels in the leads 3 and 4, we get backto the result recently obtained in Ref. [17] for a chaoticquantum dot connected to two leads. Note that the re- sult in the limit k B T, eV ≫ eV can be obtained withthe exchange of indices 1 ↔ ↔ k B T ≫ eV , eV . In this regime, we get to the thermalor Johnson-Nyquist noise, which appears in electroniccircuits at high temperatures. In this case we get back to S ( w ) / ( k B T G ) Two-terminal Three-terminal Four-terminal -20 -10 0 10 200.000.020.04 d [ S ( w ) / ( k B T G ) ]/ d ( e V ) eV FIG. 2: The upper panel shows the noise in the regime of finitefrequency with capacitive interaction, in terms of βeV . Thelower panel shows the corresponding second derivative, andthe inset depicts its behavior in the case of two terminals.Here we have set βeV = 10, β ¯ hw = 0 . τ / ¯ hβ = 1. the usual relation between noise and conductance, S ik =4 k B T G ik , where G ik stands for the conductance betweenthe leads i and k . Here the expression (6) simplifies to S ik (0) = 4 k B T e h N i ( δ ik M − N k ) M . (7)Let us now analyze the limit eV ≫ k B T , which leadsus to the relevant regime, where the shot noise power isrelated to the time-dependent fluctuations on the elec-tronic current due to the discreteness of the charge inthe regime of low temperatures, T →
0. Here we get S ik (0) = e V h N i ( δ ik M − N k ) M ( N + N )( N + N ) × (cid:20) N N ( N + N )( N + N ) (cid:21) . (8)If we take N i = N in the above result (8), we get tothe simpler result S ik (0) = e h (cid:20) δ ik − (cid:21) N V , (9)which agrees with the result given by Eq. (7) of Ref. [6],in the same limit. d [ S ( w ) / ( k B T G ) ]/ d [ e V ] S ( w ) / ( k B T G ) N =15, N =N =15, N =0 N =15, N =N =20, N =0 N =15, N =N =25, N =0 N =15, N =N =30, N =0 N =15, N =N =35, N =0 eV FIG. 3: The upper panel shows the noise in the regime of finitefrequency with capacitive interaction, in terms of βeV . Thelower panel shows the corresponding second derivative. Herewe have set βeV = 10, β ¯ hw = 0 . τ / ¯ hβ = 1. IV. FINITE FREQUENCY AND CAPACITIVEINTERACTION
Let us now study the noise in the regime of non vanish-ing frequency. As we are going to show, we will need tointroduce capacitive interaction in order to ensure chargeconservation through the chaotic quantum dot. In thissense, let us first consider the simpler case, where we takethe case of nonzero frequency.
A. The Case of Nonzero Frequency
In order to get to the noise at finite frequency, we haveto substitute (4) and (5) into Eq. (2). As before, thefinal result leads us to an awkward expression. For thisreason, we focus on some relevant regimes: firstly, weconsider the case where k B T ≫ eV , eV . We get S ik ( w ) = e h N i ( δ ik M − N k ) M coth (cid:18) ¯ hw k B T (cid:19) × hw (cid:2) δ ik w τ D M/ ( M − N k ) (cid:3) w τ D . (10)Here we note that lim w → S ik ( w ) = S ik (0), where S ik (0)is given by the previous result (7), as expected. We also d [ S ( w ) / ( k B T G ) ]/ d ( e V ) S ( w ) / ( k B T G ) N =N =N =N =15 N =N =15, N =N =20 N =N =15, N =N =25 N =N =15, N =N =30 N =N =15, N =N =35 eV FIG. 4: The upper pannel shows the noise in the regime offinite frequency with capacitive interaction, in terms of βeV .The lower pannel shows the corresponding second derivative.Here we have set βeV = 10, β ¯ hw = 0 . τ / ¯ hβ = 1. note that after taking i = k in the result (10), one verifiesthe presence of an asymmetry in the exchange among thenumber of open channels in the leads N , N , N e N .This asymmetry in the indices of open channels amongmultiple terminals is similar to the asymmetry found inchaotic quantum dots with two terminals [20].Another case of interest is given by eV ≫ k B T, eV .Here we get S ik ( w ) = e V h N i ( δ ik M − N k ) M × δ ik w τ D M/ ( M − N k )1 + w τ D ( N + N )( N + N ) × (cid:20) N N ( N + N )( N + N ) (cid:21) . (11)We see that lim w → S ik ( w ) = S ik (0), where S ik (0) isgiven by the previous result (8), as expected. To getto the limit eV ≫ k B T, eV , in the above result (10)and (11), we change the indices as follows: 1 ↔ ↔ N , N = 0 in the above S ( w ) / ( K B T G ) eV =0 eV =5eV =10 d [ S ( w ) / ( K B T G ) ]/ d ( e V ) eV FIG. 5: Similar plots of the noise S and its second deriva-tive. Here we have set N = 15, N = 40, N = 50, N = 25, β ¯ hw = 0 . τ / ¯ hβ = 1. results (10) and (11), we get back to results obtained inRef. [20] in the case of two terminals. B. Presence of Capacitive Interaction
The noise at finite frequency, as given in (2), has thesymmetry S ki ( w ) = S ik ( − w ), but it does not ensurecurrent conservation [1]. To remedy the issue and en-sure current conservation, according to [7] we have totake into account the displacement current, induced bythe contacts and gates included in the chaotic quantumdot, as shown in Fig. 1. We follow this line, and so wesubstitute the matrix A βα ( ε, ε ′ ) by the effective matrix A βα ( ε, ε ′ ) + ∆ A βα ( ε, ε ′ ), which is defined as the currentmatrix plus the displacement current matrix, into theequation (2). The displacement current introduces thecontribution [7]∆ A βα ( ε, ε ′ ) = 2 πi ¯ hwG α ( w ) × (cid:20) i π β S † ( ε )1 α ( S ( ε ) − S ( ε ′ )) ε − ε ′ (cid:21) , where G α ( w ) = N α M (1 − iwτ ) τ D τ C + τ D , is the frequency dependent response function, which waspreviously obtained in [20], in the case of two terminals,with τ being the charge relaxation time induced by thedisplacement current. The charge relaxation time is de-fined in terms of the dwell time, τ D , and the RC time, τ C = hC/ ( M e ), where C is the geometric capacitanceof the chaotic quantum dot. It obeys τ − = τ − D + τ − C .To obtain the noise in terms of the charge relaxationtime, we now have to calculate the average of the trace of( A αβ ( ε, ε ′ ) + ∆ A αβ ( ε, ε ′ )) ( A βγ ( ε, ε ′ ) + ∆ A βγ ( ε, ε ′ )) . (12)We substitute the corresponding result into (2) to get toexact expression for the noise power in the case of finitefrequency with capacitive interaction. In the limit of hightemperature, k B T ≫ eV , eV , we obtain the followingresult S ik ( w ) = e h N i ( δ ik M − N k ) M coth (cid:18) ¯ hw k B T (cid:19) hw (cid:2) δ ik w τ M/ ( M − N k ) (cid:3) w τ (13)Also, in the limit eV ≫ k B T, eV , we get S ik ( w ) = e V h N i ( δ ik M − N k ) M δ ik w τ M/ ( M − N i )1 + w τ ( N + N )( N + N ) (cid:20) N N ( N + N )( N + N ) (cid:21) (14)We note that the above results (13) and (14) depend onthe charge relaxation time, τ . Thus, we can now ensurethat the noise satisfies current conservation through thechaotic quantum dot. It is interesting to note that ifwe remove the capacitive environment, that is, if we set τ C = 0, we get back to the previous results (10) and(11). We also note that the limit eV ≫ k B T, eV canbe obtained from the result (14), with the index changes:1 ↔ ↔ V. PHENOMENOLOGY OF THE NOISE ANDASYMMETRIES INDUCED BY THE EXTRATERMINALS
The evaluation of the main term of the noise in achaotic quantum dot, done with the above generalityleads us to large analytical expression of low intuitive ca-pability. To circumvent problem with the understandingof the large analytical expressions, in the current Sec-tion we will investigate generic possibilities by meansof graphical descriptions, which we believe expose thebehavior of the chaotic quantum dot much more sig-nificantly. As we shall show, the chaotic quantum dotpresents a very rich phenomenology, generated by thecompetition among the number of terminals, the tensionin the reservoirs and the field of finite frequency. We start with the investigation of the noise as a func-tion of the difference of potential between terminals 1 and2, βeV . This is depicted in Fig. 2, where we used thetypical values N = 15, N = 40, βeV = 10, β ¯ hw = 0 . τ / ¯ hβ = 1. When the number of open channels in ter-minals 3 and 4 are set to zero, N = N = 0, we see thatthe noise power is symmetric, with respect to βeV . Thisresult corresponds to the system with two terminals, andit is shown in the inset in Fig. 2. However, we observean asymmetry when we open channels in the leads 3 and4. We also note that this asymmetry depends on thenumber of terminals, but it is still present when lead 4is closed. To highlight the effect, we also plot the secondderivative of the noise power. The plots clearly show thepresence of two secondary peaks, also asymmetric.In Fig. 3, we show the symmetry breaking of the noisepower, in terms of the difference of potential in terminals3 and 4. There we also show that the symmetry breakingoccurs as we vary the number of open channels in theterminals. When the number of open channels in theterminals are symmetric, that is, when one sets N = N = N , there is no asymmetry, even though we keepterminal 4 closed. In a similar way, we show in Fig. 4 thatan asymmetry is also present when there is asymmetryin the number of open channels, in the case with all fourleads operating.In Fig. 5 we depict competition between V and V .We note that the second derivative shows the presence ofsecondary peaks, which increase for increasing V andare centered exactly at values where V equals V , or S ( w ) / ( K B T G ) hw=0.2, eV =10 hw=0.5, eV =10 hw=1.0, eV =10 d [ S ( w ) / ( K B T G ) ]/ d ( e V ) eV FIG. 6: Plots of the noise S and its second derivative. Wenote a suppression in the peaks of the second derivative of thenoise due to the increasing of the frequency. Here we have set N = 15, N = 40, N = 50, N = 25, βeV = 10 and τ / ¯ hβ = 1. V . This fact reflects the change of the behavior of thenoise due the competition between the potentials. Wealso note that the noise is symmetric when V or V vanishes, and that the asymmetry appears only whenthey are nonzero, and different from each other.We have also analyzed the behavior of the noise fordifferent values of β ¯ hw . We illustrate this case in Fig. 6,where we show that the increasing of β ¯ hw tends to sup-press the noise, but the asymmetry due to the potentialis still effective.Finally, in Fig. 7 we depict the noise in terms of thenumber of open channels in the leads. We set N = N , N = N , βeV = 10, β ¯ hw = 0 . τ / ¯ hβ = 1, in orderto obtain for S ( βeV = 15) the approximate value10(259 N + 1265 N N + 1023 N N + 478 N )1030 N + 4130 N N + 5160 N N + 2063 N , (15)and for S ( βeV = − N + 7110 N N + 4568 N N + 1017 N N + 1808 N N + 2259 N N + 304 N . (16)In Fig. 7 we depict expressions (15) and (16) in termsof N = N . They behave differently, and the difference
10 15 20 25 30 35 402.22.32.42.5 S ( w ) / ( k B T G ) N N =N =15, N =N , eV = 15 N =N =15, N =N , eV = - 15 N =N =25, N =N , eV = 15 N =N =25, N =N , eV = - 15 FIG. 7: Noise in the regime of finite frequency with capaci-tive interaction, in terms of the number of open channels interminals 2 and 3. Here we have set βeV = 10, βeV = 15, β ¯ hw = 0 . τ / ¯ hβ = 1. induces the asymmetry of the noise in terms of the num-ber of open channels in the leads. We also note that thetwo expressions are equal for N = N , meaning that N = N removes the asymmetry. VI. COMMENTS AND CONCLUSIONS
In this work we studied a chaotic quantum dot con-nected to four leads in several distinct limits, focusingon the noise at finite frequency and temperature, in thenoninteracting and interacting regimes. We used the di-agrammatic procedure to obtain analytic results for themain term of the semiclassical expansion for the noise bymeans of the average over the Wigner-Dyson ensembles.With the diagrammatic procedure, we could write gen-eral expressions for the noise. In particular, we studiedthe noise at lower (shot noise power) and higher (Nyquist-Jonhson noise) temperatures, obtaining general expres-sions at vanishing or nonvanishing frequency. We alsoinvestigated the interacting and non interaction cases,using the capacitive environment under the presence ofan AC frequency in the terminals. The main resultsshow the existence of surprising effects induced by thesubtle combination of spatial and temporal phase coher-ence in the electronic modes in the chaotic quantum dot.Another important issue concerned the topology of themulti-terminal chaotic cavity, which is crucial to mod-ify the behavior of the noise, such as the suppression ofthe peaks of its second derivative, and the presence ofasymmetry due to nonvanishing tensions applied in thequantum dot. The results also show the importance ofthe number of open channels in the terminals, to con-trol the behavior of the noise in a multi-terminal chaoticcavity with arbitrary topology.The study is of current interest and may contributeto understand future experiments on mesoscopic systemswith four leads. The effect of topology does not neces-sarily leads to asymmetry in the conductance, as one cansee from the study of Ref. [25]. However, the quantumnature of the noise in mesoscopic structures appears in adirect manner in the topology. In the very recent work[26], the authors experimentally identify an asymmetry inthe noise, induced by the temperature decreasing belowthe Kondo temperature. The results of the present workare also valid at higher temperatures, so that the asym-metry here identified could be observed in the crossoverbetween the thermal or Nyquist-Johnson noise and theshot noise.To make the current results stronger, we also investi-gated the chaotic cavity with six, eight and ten terminals,and the presence of two, four and six extra terminals does not destroy the asymmetry observed in the case of fourterminals. We believe that the symmetry present in thecase of two terminals is specific to such simple topology,and we hope that our results contribute to the experimen-tal control of the noise in multi-terminal chaotic cavities.There are much more to be done, and we are presentlyinvestigating the case of a non ideal chaotic cavity, in-cluding the crossover between the orthogonal and uni-tary ensembles in the weak localization correction, andthe role played by the spin accumulation effect in ferro-magnetic structures.
Acknowledgments
This work was partially supported by the Brazilianagencies CAPES, CNPq and FACEPE-PE. [1] Ya. M. Blanter and M. B¨uttiker, Phys. Rep. , 1(2000).[2] S. Datta,
Electronic Transport in Mesoscopic Systems (Cambridge University Press, 2001).[3] Pier. A. Mello and Narendra Kumar,
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