On the single-particle-reduced entropy of a gated nanowire system in the Coulomb blockade regime
José María Castelo, Klaus Michael Indlekofer, Jörg Malindretos
OOn the single-particle-reduced entropy of a gated nanowiresystem in the Coulomb blockade regime
Jos´e Mar´ıa Castelo and Klaus Michael Indlekofer ∗ RheinMain University of Applied Sciences,FB ING / Institute of Microtechnologies,Am Br¨uckweg 26, D-65428, R¨usselsheim, Germany
J¨org Malindretos
Georg-August-Universit¨at G¨ottingen,IV. Physikalisches Institut, D-37077, G¨ottingen, Germany (Dated: May 29, 2013)
Abstract
In this paper, the single-particle-reduced entropy of a nanowire field-effect transistor (NWFET)in the Coulomb blockade regime is studied by means of a multi-configurational self-consistentGreen’s function approach. Assuming that the many-body statistical preparation of the systemis described by a mixture of Slater determinants of relevant natural orbitals, the single-particle-reduced entropy can be interpreted as a measure of the degree of mixture of the system’s prepara-tion. Considering the realistic case of an InP based NWFET, we present current–voltage charac-teristics and entropy diagrams for a range of equilibrium and non-equilibrium states. Signaturesof few-electron Coulomb charging effects can be identified, as known from experimental situations.Furthermore, we illustrate the significance of the single-particle-reduced entropy by analyzing thecorresponding electronic configurations. ∗ Electronic address: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] J un . INTRODUCTION Semiconductor nanowires have attracted great interest as candidates for nanoelectronicapplications and might become a basis for new and high-performance devices on the nanome-ter scale [1, 2]. They enable ultimately scaled conventional FETs as well as novel transistorconcepts, which are expected to offer superior performence in terms of current drive ca-pability and subthreshold slope [3]. Moreover, nanowire based transistor devices allow forthe investigation of fundamental questions of electronic transport in quasi one-dimensionalquantum systems.A manifold of numerical studies exists that approaches the subject of electronic transportin NWFETs using quantum kinetic methods for application-relevant conditions, based on anon-equilibrium Green’s function (NEGF) formalism [4–9]. However, one of the challenges inthe realistic simulation of nanodevices consists in the large number of degrees of freedom (e.g.localized orbitals) in combination with many-body Coulomb interaction effects, which makea full Fock-space description numerically unfeasible. Therefore, the majority of nanodevicesimulation techniques on realistic length scales are based on a mean-field approximationof the Coulomb interaction. In this context, the multi-configurational Green’s functionapproach [10, 11] combines the numerical advantages of a mean-field approximation witha Fock-space description of a small sub-set of relevant, resonantly trapped states whichare responsible for Coulomb charging effects due to individual electrons. In this paper, weanalyze the role of these relevant states and the resulting configurations by means of thesingle-particle-reduced entropy [12, 13] for the case of a realistic InP based nanowire system.The structure of the paper is as follows. First, we briefly describe the employed quantumkinetic simulation approach, followed by a definition of the considered nanodevice. Then insection IV, we focus on the electronic transport in the Coulomb blockade regime and showsignatures of few-electron Coulomb charging effects. Section V is devoted to the analysis ofthe single-particle-reduced entropy of the system. Finally, the conclusions are drawn.
II. SIMULATION APPROACH
For our calculations, we have employed a multi-configurational self-consistent Green’sfunction (MCSCG) approach [10, 11] for a realistic description of discrete resonant states,2IG. 1: Schematic view of the NWFET geometry, where d ox and d ch are the gate insulatorthickness and the channel diameter, respectively. (a) Absolute value of the drain current | I D | (nA). (b) Single-particle-reduced entropy S (bit). FIG. 2: Simulated current–voltage characteristics (a) and single-particle-reduced entropy(b).beyond the mean-field approximation. We make use of a one-band tight-binding descriptionof the channel in the effective mass approximation, represented by a basis of 2 N sites localized1D single-particle spin orbitals. The benefit of the MCSCG approach stems from the self-consistent division of the channel system into a small subsystem of resonantly trapped(relevant) states for which a many-body Fock-space treatment becomes numerically feasible,and the rest of the system which is treated adequately on a conventional mean-field level. The3ock-space description allows for the calculation of few-electron Coulomb charging effectsbeyond mean-field, as manifested in the Coulomb blockade regime.In the following, ρ denotes the single-particle density matrix [14] of the system undernon-equilibrium conditions. For the NWFET channel, ρ is a 2 N sites × N sites matrix and isobtained from the NEGF formalism. The single-particle eigenvectors of ρ are called naturalorbitals [14], its eigenvalues are real and within the interval [0 , ρ would have only eigenvalues 0 and 1.The MCSCG approach describes the state of the channel with fluctuating electron numberas a weighted mixture of a few relevant many-body states, so-called configurations . Inthe current approximation, these configurations are assumed to be Slater determinants ofresonantly trapped natural orbitals [15]. The corresponding weights are real, restricted tothe [0 ,
1] interval, and their sum is unity. Thus, the relevant many-body statistical operatorconsists of a weighted sum of projectors, restricted in our case to a relevant sub-space of thewhole Fock-space.
III. DEVICE LAYOUT
Figure 1 shows the elements of the considered coaxially-gated NWFET. The channelconsists of an InP nanowire, which has a lattice constant a = 5 . m ∗ /m e = 0 .
079 and a relative dielectric constant (cid:15) r = 12 . N sites = 30 giving a length L = 17 . d ch = 5 nm.The channel and the gate are separated by a SiO oxide layer of thickness d ox = 10 nmand relative dielectric constant (cid:15) r = 3 .
9. The source and drain are contacted with Pd/Ti toform Schottky barriers [17] of height Φ SB = 0 . T = 77 K. IV. SIMULATED ELECTRONIC TRANSPORT
At low enough temperatures and drain–source voltages V DS (compared to the channel’sCoulomb charging energy and single-particle energy level spacing), the energy which isnecessary to add an extra electron to the channel can exceed the thermal energy, and the4 oint N e Shell ( V GS , V DS ) S Slater determinants WeightsA: Empty channel 0 Empty (0.034, 0) 0.074 (00000000) 1B: 1st Diamond 1 Open (0.15, 0) 1.07 (10000000) 0.502(01000000) 0.498C: Non-equilibrium state (0.23, 0.026) 0.908 (10000000) 0.363(01000000) 0.361(11000000) 0.276D: 2nd Diamond 2 Closed (0.285, 0) 0.079 (11000000) 1E: 3rd Diamond 3 Open (0.405, 0) 1.102 (11000000) 0.024(11100000) 0.489(11010000) 0.487
TABLE I: Data from selected points in the diagrams. The table shows the electron number N e , shell filling, voltage coordinates ( V GS , V DS ), single-particle-reduced entropy S , thedominant Slater determinants and their associated weights { w i } .current through the NWFET is blocked. Within this Coulomb blockade regime, the NWFETbehaves as a single-electron transistor (SET) [18–20].We have simulated electronic transport in this regime by means of the MCSCG approach.Coulomb diamonds emerge in the current–voltage characteristics, as shown in Fig. 2a. Inthese regions, the drain current I D reduces to almost zero by Coulomb blockade and thechannel is occupied by an integer number of electrons N e . Along the V DS (cid:39) N e and N e + 1. By varying the gate–source voltage V GS , Coulomb oscillations can be observed. Thesimulations correctly resemble the features that are observed in experiments [21]. V. SINGLE-PARTICLE-REDUCED ENTROPY
The single-particle-reduced entropy in bit is defined as [12, 13] S ≡ − Tr( ρ log ρ ) . (1)5n quantum chemistry, S is referred to as the correlation entropy [12, 22] for pure many-body states. In general, S indicates the deviation from a single Slater determinant, dueto correlation and/or mixture. Expanded in terms of the natural orbital basis, S canbe written as S = − (cid:80) i n i log n i , where n i denote the eigenvalues of ρ . In this paper,the NWFET channel is described by a mixture of a few relevant Slater determinants withdetailed Coulomb interaction, whereas the ”non-relevant” rest is treated on a mean-fieldlevel. Then, S measures the degree of mixture of the system’s preparation.In the extreme case of a many-body preparation that is a mixture of Slater determi-nants with a set of fully disjoint single-particle states (i.e., with complementary occupa-tion), N SD = 2 S /N fe can be interpreted as the number of relevant Slater determinants in thepreparation [13], where N fe denotes the number of fluctuating electrons. For example, pointB in Table I corresponds to N fe = 1.Figure 2 shows the simulated characteristics which clearly exhibit a correspondence be-tween the Coulomb diamonds in the current–voltage characteristics and certain diamond-likeshaped structures in the entropy diagram. For those regions where the current is blockedand the channel occupied by 0 , , , . . . electrons (Coulomb diamonds from left to right) thestate of the device jumps consecutively from open shell ( N e = 1 , . . . and multiple Slaterdeterminants) to closed shell ( N e = 2 , . . . and a single Slater determinant). S indeedreflects these mixtures and therefore its geometry is also diamond-shaped. This will be seenin detail in the following.Table I gathers information obtained by means of the MCSCG formalism, correspondingto different voltage points in Fig. 2. Point A is associated with an empty channel ( N e = 0),whose state is given by the vacuum state with unity weight. S (cid:39) N e = 1). Here, the single electron in thechannel ( N fe = 1) has the chance to occupy one of the two states in the first single-particleenergy level: with spin up or with spin down (open shell configuration). This is reflectedin the two Slater determinants ( N SD (cid:39)
2) with almost equal weights w (cid:39) w (cid:39) . S (cid:39) w (cid:39) w (cid:39) w (cid:39) .
3. This situation shows that it is equally probable in this state to6IG. 3: Electron number N e and single-particle-reduced entropy S (bit) as a function of V GS for fixed V DS = 0.find an electron in the first single-particle energy level with spin up or down, or two electronswith both spin directions.Point D corresponds to the second diamond ( N e = 2). Each electron occupies the firstsingle-particle energy level with spin up and down respectively ( N fe = 0). Therefore, thereis only one Slater determinant to describe this situation and so its weight is unity. S (cid:39) N e = 3), very similar to that of thefirst diamond. Here, two electrons occupy the first single-particle energy level, and a thirdone ( N fe = 1) has both possibilities of spin occupation in the second single-particle energylevel. Although there is a small contribution from a doubly occupied Slater determinant,we can see that N SD (cid:39)
2, and the last two determinants are the main contributors ( w (cid:39) w (cid:39) w (cid:39) . S is slightly higher than one reflects that the first determinantalso plays a minor role.All of the near-equilibrium cases corresponding to Coulomb diamonds that we have an-alyzed can be summarized in Fig. 3, in which N e and S are plotted against V GS for fixed7 DS = 0. The N e curve exhibits integer charging steps. On the other hand, S oscillatesapproximately between zero and one. It can be seen here that S (cid:39) N e (cid:39) S (cid:39) N e (cid:39) S , leading to slightly increasedvalues. VI. CONCLUSION
We have simulated quantum transport properties of a realistic gated nanowire system inthe Coulomb blockade regime by means of a multi-configurational Green’s function approach.The current–voltage characteristics correctly resemble the experimentally known signaturesof few-electron Coulomb charging effects. In addition, the single-particle-reduced entropyhas been analyzed for several channel states, showing that it is a measure of the mixture inthe system’s thermodynamical preparation. In this sense, S can be interpreted as the lackof information about the Slater determinant in which the system can be found. VII. ACKNOWLEDGEMENT
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