Statistical analysis of the transmission based on the DMPK equation: An application to Pb nano-contacts
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Statistical analysis of the transmission based on the DMPKequation: An application to Pb nano-contacts
V´ıctor A. Gopar , Instituto de Biocomputaci´on y F´ısica de Sistemas Complejos, Corona de Arag´on 42, 50009 Zaragoza, Spain Departamento de F´ısica Te´orica, Facultad de Ciencias, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spainthe date of receipt and acceptance should be inserted later
Abstract.
The density of the transmission eigenvalues of Pb nano-contacts has been estimated recentlyin mechanically controllable break-junction experiments. Motivated by these experimental analyses, herewe study the evolution of the density of the transmission eigenvalues with the disorder strength and thenumber of channels supported by the ballistic constriction of a quantum point contact in the frameworkof the Dorokhov-Mello-Pereyra-Kumar equation. We find that the transmission density evolves rapidlyinto the density in the diffusive metallic regime as the number of channels N c of the constriction increase.Therefore, the transmission density distribution for a few N c channels comes close to the known bimodaldensity distribution in the metallic limit. This is in agreement with the experimental statistical-studies inPb nano-contacts. For the two analyzed cases, we show that the experimental densities are seen to be welldescribed by the corresponding theoretical results. PACS.
Since first transport experiments in two-dimensional elec-tron gases confined in semiconductor heterostructures witha constriction comparable to the Fermi wave length –quantum point contacts– where conductance quantizationwere reported [1,2], much attention has been given todescribe theoretically the quantum electron transport inthese kind of systems. The dimensions of those structuresare small enough that quantum interference effects can beobserved in transport measurements, i.e., the typical sizeof the systems is smaller than the phase-coherent length.More recently, scanning tunneling microscope and me-chanically controllable break-junction (MCBJ) techniqueshave made possible the fabrication of structures at theatomic scale. We refer to the review article [3] for a de-tailed description of the progresses on those experimentaltechniques. In particular, in MCBJ experiments a notchedwire is elongated with high mechanical precision and sta-bility; thus, during this process atomic-scale contacts, orquantum point contacts, are formed between two wideelectrodes.From the theoretical side, several aspects of the trans-port in quantum point contacts have been investigatedand an extensive literature already exists. We refer thereader to [3] and [4] for a review of the topic. Here wemention some examples of the problems addressed in theliterature where random configuration of impurities in thesystem is an important ingredient and therefore a statis- tical study of the transport is relevant. For instance, thesuppression of the conductance fluctuations due to disor-dered reservoirs attached to a ballistic constriction wasstudied early in the nineties [5,6]. Statistical properties ofthe conductance, resistance, and shot noise in disorderedwires with a constriction were studied in Ref. [6]. More re-cently, effects of impurities around a quantum point con-tact on the statistics of the transmission eigenvalues havebeen investigated [7]. In single-atom contacts, it has beenconjectured and experimentally verified that the numberof conducting channels is determined by the valence or-bitals of the constriction atom [3,8].Lately, several aspects of the electron transport in quan-tum point contacts have been experimentally analyzedusing MCBJ techniques [3,9,10]. For example, statisticalproperties of the transport like the density of the transmis-sion eigenvalues have been estimated for Pb nano-contacts[10,11]; in these experiments, it has been pointed out thatthe transmission density for contacts with a few openchannels is unexpectedly similar to the known bimodaldensity distribution in the diffusive metallic regime. Werecall that in the metallic limit a large number of chan-nels are assumed to contribute to the transport.Motivated by recent break-junctions experiments wheresome statistical properties of the transmission eigenval-ues have been estimated for Pb nano-contacts [10,11], inthis paper we calculate the density of the transmissioneigenvalues within the framework of the Dorokhov-Mello-
V. A. Gopar: Statistical analysis of the transmission based on the DMPK equation: An application to nano-contacts
Pereyra-Kumar (DMPK) equation [12,13]. In order to ap-ply DMPK ideas to those break-junctions experiments,we assume that the constriction, or neck, formed duringthe elongation processes of the MBCJ device is smalleror of the order of the mean free path. This nowadays isachieved experimentally; in fact, the narrowest part of aMBCJ can be of molecular-size scale. We will considerthat the random character of the electronic transport isdue to the disordered wide regions at both sides of thecontact. Finally, we suppose that the neck and the widedisordered region of the MBCJ device have a step geome-try (as in Fig. 1, upper sketch). Thus, our calculations arebased on the the solution of the DMPK equation given inRefs. [14,15,16] and a mapping of the transport problembetween constricted and unconstricted wire geometries in-troduced in Ref. [6]. We will show that in agreement withthe results reported in Ref. [10], our theoretical calcula-tions show that the transmission density can be close tothat one in the diffusive metallic limit, even when theballistic constriction of the point contact supports a fewchannels.In subsection 1.1, we introduce the DMPK equationand its solution for a single and multiple open channels.In the same subsection, we present the above-mentionedmapping of the problem of a wire with a constriction tothat one without such constriction. In Sec. 2, we introducethe functions of the transmission density that we study inthis work. Our analysis of the density start in subsection2.1 with the simplest case of a point contact with a ballisticconstriction supporting one channel. In subsection 2.2, weanalyze the case when several open channels are supportedin the ballistic constriction. Also we compare some of theresults for the transmission density reported in Ref. [10]with our calculations. Finally, we give a summary of ourstudy in Section 3.
A scaling equation for the transmission eigenvalues { τ i } of a quasi-one-dimensional disordered system, or quan-tum wire, has been developed in the past [12,13]. Withinthis framework, the evolution of the distribution of thetransmission eigenvalues p ( { τ i } ) [or equivalently p ( { λ i } ),where λ i = (1 − τ i ) /τ i ] with the length L of the quantumwire is given by the Fokker-Planck equation l ∂p ( λ ) ∂L = 2 N + 1 1 J ( λ ) × X a ∂∂λ a (cid:20) λ a (1 + λ a ) J ( λ ) ∂p ( λ ) ∂λ a (cid:21) , (1)known as DMPK equation, where N is the number ofchannels, or transverse modes, and l is the mean free path,while J ( λ ) = Q Ni 11 + λ i , (3)where g is the dimensionless conductance. Thus, the sta-tistical properties of g are governed by that ones of thetransmission eigenvalues.The solution of the DMPK equation is known also forthe general case of multiple channels and for any sym-metry ( β = 1 , , 4) [14,18]. The general solution, unfortu-nately, is quite complicated, but it simplifies in the metal-lic (1 ≪ s ≪ N ) and insulating ( L >> N l ) regimes. Inthese two opposite regimes of transport, the solution ofthe DMPK equation can be written in the following form p ( λ ) = 1 Z exp[ − H ( λ )] , (4)where Z is a normalizing factor, and H ( λ ) is given by H ( λ ) = N X i 12 ln | sinh x i − sinh x j | − 12 ln | x i − x j | ,V ( x i ) = ( N + 1)2 s x i − 12 ln | x i sinh 2 x i | , (6)while in the insulating regime u ins ( x i , x j ) = − 12 ln | sinh x i − sinh x j | − ln | x i − x j | ,V ins ( x i ) = ( N + 1)2 s x i − 12 ln | x i sinh 2 x i | − 12 ln x i (7)In writing Eqs. (6) and (7) we have assumed the presenceof time-reversal invariance in the system ( β = 1). We notethat Eq. (7) differs from Eq. (6) only in the ln | x i − x j | andln x i terms. We remark that in the metallic regime a largenumber of channels, or x i variables, contribute to the con-ductance, whereas in the insulating regime, the main con-tribution to the conductance comes from one (the smallest . A. 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The upper sketch represents a quantum wire withmean free path l and a ballistic constriction in the middle.The wire can support N channels. As explained in the text,the DMPK equation for such kind of wire can be mapped ontoa DMPK equation describing a wire (as the one represented inthe lower sketch) without the constriction, mean free path l ′ ,and number of channels N c . one) x i . On the other hand, in the insulating regime, theaverage conductance is very small and x i >> 1; in thislimit, ln | x i − x j | and ln x i are negligible compared to theother terms in the insulating case Eq. (7). Since thesetwo terms are the only ones that differ from the metallicsolution, we thus assume that the metallic solution canbe used as an approximated solution to DMPK equationvalid beyond the metallic limit and for a finite number ofchannels.Finally, we recall that in the diffusive metallic regime,the density of the transmission eigenvalues defined by theaverage h P i δ ( τ − τ i ) i is known to follows the bimodaldistribution [4,19]: ρ met ( τ ) = h g i τ √ − τ , (8)with h g i = N l/L . As mentioned in the previous subsection, the DMPK equa-tion was derived for a wire geometry, i.e., a quasi-one-dimensional system of length L and width W . On theother hand, however, in MCBJ techniques a notched wireis bended to produce a small constriction between twowide electrodes. This constriction can be of the order ofthe mean free path of the sample, i.e., a ballistic con-striction. The problem of transport through a quasi-one-dimensional wire with a constriction can be mapped on toa problem of a quantum wire without the constriction, asit was shown by Beenakker and Melsen [6]: it turns outthat the DMPK equation for a wire with a ballistic con-striction with mean free path l and N number of channels(Fig. 1, upper wire) is equivalent to that one for a wire ge-ometry (without the constriction, Fig. 1, lower wire) withmean free path l ′ given by l ′ = l ( N + 1) / ( N c + 1) , (9)where N c is the number of channels determined by theballistic constriction in the wire. We note that a step-constriction geometry is assumed in the model. This map-ping allows us to use the known results for unconstricted τ ρ no r m ( τ ) τ ρ ( τ ) Fig. 2. (Color on-line) Density of the transmission eigenval-ues as defined in Eq. (10) for Pb nano-contacts with conduc-tances (in units of e /h ) g = 4 (solid upward triangles), 6 (solidsquares), 10 (solid circles), and 15 (solid downward triangles).The solid (black) line is the bimodal distribution given by Eq.(8). The inset shows the density ρ ( τ ) for the same values of theconductances of the main frame. The data points for ρ norm ( τ )and ρ ( τ ) have been obtained from Refs. [10] and [11]. wire geometries, in particular the solution Eq. (4-6), tostudy the statistical properties of transport in quantumpoint contacts; we will be especially interested in func-tions of the transmission density introduced in Ref. [10]. The density of the transmission eigenvalues ρ ( τ ) were es-timated in Ref. [10] by analyzing several samples of Pbnano-contacts. To be specific, the functions ρ norm ( τ ) and q ( τ ) defined as ρ norm ( τ ) = 1 h g i ρ ( τ ) , (10)and q ( τ ) = 1 h g i Z τ dτ ′ τ ′ ρ ( τ ′ ) , (11)were introduced by Riquelme et. al. [10,11]. We find con-venient and motivating to reproduce here some of the re-sults of Refs. [10] and [11]. In figure 2, we show ρ norm ( τ )and ρ ( τ ) for Pb contacts with conductances ranging from4 to 15 (in units of the conductance quantum 2 e /h ) [20].We can observe in Fig. 2 that ρ norm ( τ ) comes close to themetallic limit result [solid line, Eq. (8)], even for contactswith small conductances. This was the unexpected resultpointed out by the authors of Ref. [10].Coming back to the theoretical model, according tothe mapping introduced above and assuming that N c isthe number of channels determined by the constriction in V. A. Gopar: Statistical analysis of the transmission based on the DMPK equation: An application to nano-contacts the wire, the transmission density is given by ρ ( τ ) = * N c X i δ ( τ − τ i ) + . (12) We start with the simplest case of a quantum point contactwith one open channel ( N c = 1). It is instructive to startwith this case, although no experimental data are availableto compare to. Using that τ = 1 / (1 + λ ) in Eq. (2), weimmediately obtain for the density: ρ ( τ ) = 1 √ π (cid:16) ξ L (cid:17) e − L/ (2 ξ ) τ Z ∞ y dy y e − ξy / L p cosh y + 1 − /τ , (13)where y = arccosh(2 /τ − ξ = ( N + 1) l in Eq. (13).Performing numerically the integral in Eq. (13), we ob-tain the density ρ norm ( τ ) and the cumulative function q ( τ )given by Eqs. (10) and (11), respectively. We note that forthe single channel case g = τ . In Fig. 3, we show the evo-lution of ρ norm ( τ ) as the value of the disorder parameter L/ξ is changed. As expected for L > ξ , small values of τ ,or conductance g , are favored, while as the ratio L/ξ is de-creased, the density is accumulated at values of τ near 1.As a reference, we have included the bimodal distribution(solid line) in Fig. 3. The function q ( τ ) is plotted in the in-set of figure 3 for the same values of L/ξ in the main frame.From Eq. (8), in the metallic limit: q met ( τ ) = 1 − √ − τ (solid line in the inset of Fig. 3). In all the cases, ρ norm ( τ )and q ( τ ) clearly exhibit discrepancies with respect to thediffusive metallic limit, in contrast to the results for themultichannel case, as we will see in the next subsection. Let us now assume that the ballistic constriction can sup-port several channels N c . Therefore from the expressionfor the density [Eq. (12)] and using the join probability p ( { x i } ) given by Eqs. (4 - 6), we can write the density ofthe transmission eigenvalues as ρ ( τ ) ∝ e − ξ L x p x sinh (2 x )2 τ √ − τ Z N c Y i dx i × exp − X 12 ln( x i sinh 2 x i ) . (15)We will see that the density of the eigenvalues τ n givenby Eq. (14) approximates to the diffusive metallic regime[Eq. (8)] even for a relative small number of channels N c . τ ρ no r m ( τ ) s*nu=2, Main frame: Density ρ norm ( τ ) for N c = 1 at dis-order parameter 2 L/ξ =2 (dashed line), 2/5 (dotted line),1/5 (dashed-dotted line) with average conductance h g i =0 . , . , . 83, respectively. The solid line is computed fromEq. (8) and corresponds to the diffusive metallic regime. Inset:The cumulative function q ( τ ) given by Eq. (11) for the samevalues of the parameters 2 L/ξ as in the main frame. τ ρ no r m ( τ ) N c =3N c =4 N c =6 τ q ( τ ) Fig. 4. (Color on-line) Density ρ norm ( τ ) for N c = 3 , , and6 (dotted red, dashed green, dashed-dotted blue lines, respec-tively). In the inset q ( τ ) is shown for the same values of N c in the main frame. For all cases, the average transmission h τ i is approximately ( h τ i = 0 . h g i = 1 . , , . N c = 3 , , and 6, respectively. The solid(black) line is the bimodal distribution in the metallic limit,Eq. (8). We next present the results for ρ norm ( τ ) and q ( t ) [Eqs.(10) and (11)] obtained from the above expression for ρ ( τ )[Eq. (14)], where the integrals were performed numerically.Let us start showing the evolution of the density ofeigenvalues with the number of channels N c . In figure 4, ρ norm ( τ ) and q ( t ) are plotted for N c = 3 , , and 6; these . A. Gopar: Statistical analysis of the transmission based on the DMPK equation: An application to nano-contacts 5 τ ρ no r m ( τ ) L/ ξ = 1/6 L/ ξ = 1/2 L/ ξ = 2/3 τ q ( τ ) Fig. 5. (Color-online) Evolution of the density ρ norm ( τ ) withthe disorder parameter L/ξ . In the inset, q ( τ ) is plotted forthe same values of L/ξ in the main frame. In all the cases thenumber of channels is fixed to N c = 5. The solid (black) lineis the bimodal density in the diffusive metallic regime. cases are chosen to have the same average value h τ i . Wecan observe that as the number of channels is increased, ρ norm ( τ ) concentrates at small values of τ and near 1,i.e., the density evolves to the bimodal distribution plot-ted in solid line in the same Fig. 4. In other words, asthe conductance of the ballistic constriction increases, thedistribution of the transmission eigenvalues approximatesquickly the bimodal distribution. On the other hand, thecumulative function q ( τ ) (inset in Fig. 4) also comes closeto the result in the metallic limit q met ( τ ) (solid line). Infact, we note that the different curves for q ( τ ) might lookcloser to the metallic case q met ( τ ) than the correspondingcases for ρ norm ( τ ) to ρ met( τ ) (the bimodal distribution).This is due to loss of details of the density ρ norm ( τ ) whenthis function is integrated [Eq. (11)] to obtain q ( τ ).Now we show the evolution of the density with theratio L/ξ . In figure 5, we plot the results for ρ norm ( τ )and q ( τ ) (inset) at three different values of the disorderparameter L/ξ for a fixed value of the number of chan-nels N c = 5. For the smallest value of L/ξ (dashed line), ρ norm ( τ ) shows a peak at τ ’s near 1, like the bimodal dis-tribution (solid line); in fact, the density ρ norm ( τ ) lookssimilar to this metallic limit case, except at small valuesof τ . As the ratio L/ξ is increased, i.e., going to the insu-lating regime, the peak at small values of τ grows rapidlyand ρ norm ( τ ) comes close to the bimodal distribution atsmall values of τ . An increase in peak height at small τ ’s,however, causes a detriment to the peak at τ near 1, asone might expect: roughly speaking, we can say that theevolution of ρ norm ( τ ) with the ratio L/ξ described above isconsequence of the normalization of the density function.Regardless of this behavior, for relatively small values ofthe ratio L/ξ , the main differences of ρ norm ( τ ) with re-spect to the diffusive metallic limit are seen only at smallvalues of τ , despite the small number of channels of theconstriction. τ ρ no r m ( τ ) τ ρ ( τ ) Fig. 6. (Color on-line) Comparison between the experimen-tal results (solid symbols) for the density ρ ( τ ) for Pb nano-contacts and the theoretical calculations (dashed and dashed-dotted lines). The conductances (in units of e /h ) of the ex-perimental data points are g = 6 (solid squares) and 10 (solidcircles), also shown in the inset Fig. 2. The theoretical resultsare chosen to have average values h τ i similar to the experimen-tal ones, which we have estimated from the experimental datain the region (0 . < τ < h τ i ≈ . < τ < ρ norm ( τ ) for the same cases inthe main frame. The solid (black) line is the bimodal densityin the diffusive metallic limit. Finally, we compare some of the experimental results ofRefs. [10] and [11] (Fig. 2) with theoretical predictions. Wepoint out that the theoretical model depends on the num-ber of channels N c and the disorder parameter L/ξ whichwe do not know from the experiments of Ref. [10]. Thus, inorder to make the comparison, we have extracted the av-erage transmission eigenvalue h τ i from ρ ( τ ) plotted in theinset of Fig. 2. For the couple of chosen cases, solid squaresand solid circles in Fig. 2, we have estimated: h τ i ≈ . τ are calculated in the interval of τ (0 . < τ < N c = 6 and 7 with L/ξ = 2 / h τ i similar to theexperimental ones. In Fig. 6 (main frame) we compare theresults for ρ ( τ ): the dashed line corresponds to N c = 6,while the dashed-dotted line is the result for N c = 7. Inorder to compare with the experimental results, we alsohave normalized the theoretical density to the area of theexperimental density in the interval (0 . < τ < ρ norm ( τ ) for the same cases in the mainframe. The solid line in the inset corresponds to the bi-modal distribution. V. A. Gopar: Statistical analysis of the transmission based on the DMPK equation: An application to nano-contacts We have studied the evolution of the transmission eigen-value density of quantum point contacts with the disor-der strength and the number of channels of the ballisticconstriction. Our analysis of the transmission density isbased on an approximation [15,16] to the general solu-tion of the DMPK equation and a mapping of the prob-lem of transport through a quantum wire with a ballisticconstriction to the problem of transport through a wirewithout the constriction [6] i.e., the latter problem is de-scribed by the known solution of the DMPK equation. Wehave shown that the density of the transmission eigenval-ues comes close to the known bimodal distribution in thediffusive metallic regime even for a small number of chan-nels supported by the constriction of the quantum pointcontact. This is in agreement with the unexpected resultpointed out in Ref. [10], where some statistical propertiesof the transmission eigenvalues for Pb nano-contacts wereanalyzed by using MCBJ techniques. In order to applythe DMPK approach to those experiments, we considerthat the neck formed during the elongation process in aMCBJ device is a ballistic constriction which is in betweentwo wide disordered regions. We do not have experimentalinformation of each parameter that enters in the theoret-ical model, but by comparing theoretical and experimen-tal results with similar average transmission values h τ i ,we have seen that the analyzed experimental densities arewell described by the corresponding theoretical densities.The trend of the transmission density seen in Pb contactsexperiments as the number of channels is changed is alsoin agreement with the theoretical results presented herewhen the number of channels of the ballistic constrictionis varied. I thank J. J. Riquelme for providing his Memory of Researchas well as G. Rubio-Bollinger. I also acknowledge useful corre-spondence with A. Levy Yeyati and C. Urbina. 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