Contact Block Reduction Method for Ballistic Quantum Transport with Semi-empirical sp3d5s* Tight Binding band models

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1 Purdue University Purdue e-pubs Other Nanotehnology Publiations Birk Nanotehnology Center Contat Redution Method for Ballisti Quantum Transport with Semi-empirial sp3d5s* Tight Binding band models Hoon Ryu Purdue University - Main Campus Gerhard Klimek Purdue University - Main Campus, geko@purdue.edu Follow this and additional works at: Ryu, Hoon and Klimek, Gerhard, "Contat Redution Method for Ballisti Quantum Transport with Semi-empirial sp3d5s* Tight Binding band models" (28). Other Nanotehnology Publiations. Paper This doument has been made available through Purdue e-pubs, a servie of the Purdue University Libraries. Please ontat epubs@purdue.edu for additional information.

2 Contat Redution Method for Ballisti Quantum Transport with Semi-empirial sp 3 d 5 s * Tight Binding Band Models. Hoon Ryu * and Gerhard Klimek Network for Computational Nanotehnology, Birk Nanotehnology Center, Purdue University, est Lafayette, IN 4796, USA. * ryu2@purdue.edu Abstrat The utility of the Contat Redution method (CBR) to find the retarded Green s funtion for ballisti quantum devies with semi-empirial tight binding band (TB) models is disussed. This work shows that the original method needs several modifiations to be used with TB models. In the ommon ase where two ontats are used for transport in quantum wires, our approah omputes the transmission oeffiients with muh less omputing load than the state-of-the-art Reursive Green s Funtion (RGF) algorithm.. Motivation CBR ethod, first suggested by Mamaluy et al., has reeived attention in reent years due to its ability to ompute the retarded Green s funtion for open systems with low omputing intensity. Its suessful appliation, however, was shown only with the effetive mass band (EM) model in ubi-grid bases. 2,3 The EM model works well near the ondution band minima for large devies. The quantized states, however, are not aurate if devies are on the nm-sale. 4 For the orret modeling of nano-sale devie behaviors, one therefore should use more sophistiated band models with an atomisti basis representation, whih reprodue eperimentally verified band struture of semiondutor rystals. 5 This, however, requires muh larger omputing epense. The CBR method oupled with the most sophistiate band model, therefore, may provide an eellent utility sine both quantum and atomisti effets an be properly with reasonable ompute requirements. Throughout this work, we use the semi-empirial sp 3 d 5 s* TB band model with a set of zinblende-loal (ZB) orbital bases, whih has shown its auray in estimating band strutures of various nano-sale devies and semiondutor rystals. 5,7 2. Methodology A real devie is oupled with ontats to allow arrier in-and-out flow, forming an open system desribed with a non-hermitian system matri in the non equilibrium Green s funtion (NEGF) formalism, 8 with whih we an model any quantum devies and estimate their I-V and harge profiles by omputing transmission funtions and loal density of states. The evaluation of these quantities requires the retarded Green s funtion G R (E) for an open system as a funtion of energy E as defined in (), where H o is the Hamiltonian of the losed devie and Σ is the omple self-energy term whih epresses the oupling between ontats and the devie. G R (E) = [(E + iη)i H o Σ(E)], η + () For 3D open systems, a huge omputation is required to find the partial inverse of the system matri in (). RGF signifiantly redues this numerial load, 9, however, it is still epensive and more effiient alternatives need to be onsidered, one of whih is CBR sine it redues the matri inversions to find G R. 2. CBR with Tight Binding Parameters The first step in CBR is to divide the devie spae into the boundary region that ouples with the ontats, and region d for the rest. Sine the self-energy Σ is non-zero only in the region, H o and Σ an be deomposed as (2), where the subsripts {, d} denote orresponding regions. Then G R an be omputed with the Dyson equation in (3), where we onditioned Σ and G with a Hermitian matri X to minimize matri inversions as shown in (4). H o = H o o H d o o, Σ= Σ d H d H d d d A = (I G Σ ), G R (E) = (I Σ G ) G = G d A Σ A d I d G G d G d G d (2) (3) X = d d d, Σ =Σ X, G = [EI (H o + X)] (4) = G G d G d G = Ψ α Ψ α d (η + ), E ε α + iη α where ε α,ψ α are the eigenvalues/vetors of (H o + X)

3 Here, matri inversions may not be a problem unless devies have very large boundary region, sine one only needs to invert a boundary blok A. Therefore, the major task beomes to solve eigenstates for a Hermitian matri (H o +X). The true speed-up, however, is never available without a signifiant redution in the number of used eigestates, whih an be ahieved via a smart hoie of X to make Σ negligible in the energy range of interest. 2 Devie Contat 2 L Transport Figure. Shemati piture of the semi-infinite ontat. Fig. shows the ommon approah to model ontats attahed to the devie. 2,8 The ontat is treated as a semiinfinite quantum wire of finite ross-setion where H B and represent the unit blok Hamiltonian and oupling matri between nearest bloks, respetively. Then Ψ (n,m), the eigenfuntion in the n th unit blok of the plane wave at the m th mode, will obey the Shrödinger equation and Bloh ondition in (5), where k m is the wave number of a plane-wave at the m th mode and L is the size of one unit blok along the diretion of transport. In total, there are M modes where M is a dimension of H B. (EI H B )Ψ (n, m) = + Ψ (n, m) + Ψ (n +, m), Ψ n + = ep(ik m L)Ψ n, m M (5) Now one an solve the surfae Green s funtion G surf and self-energy Σ for the ontat by formulating the general non Hermitian eigenvalue problem with (5), where their general solutions are introdued in (6), (7). 9 G surf = K[K (H B EI)K + K + KΛ] K, Σ= + G surf K = [Ψ (, ) Ψ (, 2)... Ψ (, M ) ], Λ = diag[ep(-ik L),..., ep(-ik M L)] (6) (7) The original CBR method presribes X for Σ so that (H o +X) orresponds to the devie Hamiltonian with Von Neumann boundary onditions. 2 This is feasible with the EM or k p 2 band model with ubi-grid bases, where the general epressions in (6) an be simplified to (8). G surf = KΛ K, Σ= KΛK (8) However, the simplifiation in (8) is invalid with TB models, whih use a set of atomisti bases with ZB grids. For further disussions, a simple eample is used, where we assumed two [] Si unit bloks with.2nm square ross-setion. Fig. 2 shows its oneptual shemati and orresponding devie Hamiltonian represented with the EM model (Top), and the spds* TB approah (Bottom). 2 L L2 L3 L4 L L2 L3 L4 2 Figure 2. Coneptual illustration of the eample struture and its Hamiltonian with EM (T) and TB Model (B). In the EM model, eah unit blok has a ommon grid layer suh that eah layer always has non-zero ouplings with net nearest one. However, a unit blok of [] Si has 4 unique atomi layers therefore, only the last layer in one unit blok have non-zero ouplings with the first layer in the nearest unit blok, ausing to be a singular matri as shown in Fig. 2. Then the simplifiation of (6) to (8) beomes diffiult, sine for square matries K and, K - K annot be redued to and if is singular. Therefore, one needs to evaluate the self-energy term Σ with the solution in (6) with alternatives for X to make the CBR method still pratial with TB models, one of whih we suggested in (9), where ε n is an eigenvalue at the n th sub-band minima in the ondution band (maima for the valene band) of the semi-infinite ontat. X = Σ(E = ε n)+σ(e = ε n ) + 2, for n th sub band (9) If we are interested in the arrier transport through the first few ontat sub-bands, whih is the ase to simulate RTD s or low-bias behaviors of quantum wires, the idea works very well beause the X given in (9) represents the energy-independent term of the self-energy matri at the speifi sub-band, with whih Σ beomes negligible at the viinity of the orresponding sub-band minima. 2.2 Further optimization: Cases with two ontats For the evaluation of G R, CBR needs to alulate the inverse of the boundary blok A in (3) whih assumes the ontat is oupled with the entire part of eah unit

4 blok in the boundary region of devies. In the ZB rystal strutures, however, eah unit blok onsists of a ouple of unique and epliit atomi layers suh that the requirement of matri inversion an be further simplified. To measure the numerial effiieny of the suggested method with the RGF, we assume two ontats for the arrier transport in open system, whih is the usual ase in the modeling of quantum transport. 5,9 As a detailed eample, we assume a [] Si quantum wire with 2 ontats in Fig. 3 and divide the devie spae into boundary region = + 2 and region d for the rest, where Σ in (3) is written as the epression in () sine the unit blok in the boundary region is not oupled with the one in the region 2. Then one an use () to alulate the transmission funtion T(E) with G R, whih is the boundary blok of G R and requires the inversion of a matri of size 2N for its evaluation where N represents the number of grid points in one unit blok. The size of matri inversions, however, an be further redued with the idea suggested in Fig. 4. Contat (Soure) Devie Region d 2 Contat 2 (Drain) Diretion of Transport Figure 3. Shemati piture of the eample devie. Σ = Σ 2 Σ S = Σ 2 2 Σ =Σ S +Σ D,, Σ D = T (E) = trae(γ S G R Γ D G R + ), 2 2 Σ 2 Γ S = i(σ S Σ S + ), Γ D = i(σ D Σ D + ) T (E)= trae[σ (S) G R (SD) Σ R (S) (G R (SD) )+ ], R G (SD) = (B B 2 B 3 B 4 ) (G (S) B 2B 3 G (D) ) where B = I G (S) Σ (S), B 2 = G (SD) Σ (D), B 3 = G (DS) Σ (S), B 4 = I G (S) Σ (S) () () (2) Here we deompose the boundary blok Σ, G with respet to four atomi layers in one [] Si unit blok. Then only (S) and (D) blok of Σ will beome non-zero beause only the first layer of unit blok and the last layer of unit blok 2 will be oupled with the soure and drain ontat, respetively. Then, one an easily onvert the equations for T(E) in () to (2) with simple matri arithmeti, and the evaluation of T(E) an be done with matri inversion of size N/4, whih orresponds to the number of grid points in only one of the 4 atomi layers of eah [] Si unit blok. e note the size for matri inversion now beomes 8 times smaller than the previous requirement (2N), whih results in a huge redution of omputations, as will be disussed in the net setion. S M D L L2 L3 L4 L L2 L3 L4 C at the Soure Boundary C2 at the Drain Boundary First atomi layer of C, whih is oupled with Soure ontat Last atomi layer of C2, whih is oupled with Drain ontat. Divide the boundary region into S, M, D. 2. Square matri A is deomposed as below. A = A(S) A(MS) A(DS) A(SM) A(M) A(DM) A(SD) A(MD) A(D) Figure 4. A rule to deompose boundary blok matries. 3. Result In this setion, we show the omputational pratiality of the methodology disussed so far, by investigating the tunneling behavior of eletrons through a single impurity atom plaed in the hannel of a Si quantum wire, whih is important in the tunneling spetrosopy to understand the eletroni struture of low-dimensional systems Desription of numerial eample Soure [] Transport 5nm Gate Oide nm P impurity 6.5nm 2nm (Front View) Silion 6.5nm Drain 2.3nm (Cross-Setional View) Figure 5. Shemati of the devie for simulation. A shemati of the target devie is desribed in Fig. 5. Here the wire hannel has a length of 5nm with a 2.3nm retangular ross-setion. For [] transport, we assume the soure and drain ontat with a line-doping onstant of 8 (donors/m) where the gate ontat with nm oide layer was used to onsider the band bending along the ross-setional diretion. Then a single phosphorous ion is plaed in the hannel with an analytial onsideration of the impurity potential. 4 This numerial eample uses the semi-empirial sp 3 d 5 s* TB model, 5 and the size of the orresponding devie Hamiltonian is 23,.

5 3.2 Computational effiieny of the methodology For a measurement of the numerial effiieny, the devie is simulated for 4 different ases as shown in table, where the RGF method was used as a referene. Here we only onsidered the transport of eletrons in the ondution band suh that the transmission funtion was evaluated with the first few ondution band eigenstates of (H o +X). The test has been performed with MATLAB odes on a Power Ma G5 onsisting of dual.8 GHz CPUs and 2GB of SDRAM. Table. Four different approahes for the simulation Method Number of Subdivision of Time for used ev s boundary blok sim. (s) A: CBR 3(.3%) Use 658 B: CBR (.44%) Use 647 C: CBR (.44%) Don t Use 334 D: RGF Transmission Transm ission () Peak Figure 6. Transmission profiles. (a) entire energy range (b) near the st, () 2 nd, and (d) 3 rd resonane peak..5 Integrated Tr. (a).5.5 e- e-2 e-4 (b) Peak (d) Peak 3 5e Fermi Fermi Figure 7. Integrated transmission profiles with respet to the ontat fermi-level represented with a linear (Left), and a log sale (Right). Fig. 6 shows the transmission profiles for Cases A, B, and D, with the lose-up results fousing on the first 3 resonane peaks, where the CBR method gives a result loser to the referene if one uses more eigenstates. The ase B with eigenstates reprodues the referene over almost the entire energy range of our interest. The ase A still shows a good performane at the viinity of the first resonane, but the result starts to deviate near the seond peak, and finally vanishes. The auray of results is also supported with Fig. 7, where we integrated transmission funtions with respet to the fermi-energy of ontats in equilibrium. The required times for different simulations, the details of whih were summarized in table, indiate that the alleviation of matri inversions indeed results in a huge redution of omputing epenses. 4. Summary The CBR method is shown to be pratial to ompute the retarded Greens funtion for open devies with atomisti band models. A 3D nanowire with an embedded impurity whih ehibits tunneling behaviors with etremely sharp resonanes, is used to demonstrate that the transmission an be omputed with few eigenstates of the losed system. The matri inversion needed to ompute the transmission an be redued suh that one an save signifiant omputing osts. Aknowledgments e aknowledge support from the National Siene Foundation under ontrat No. 762 and the Semiondutor Researh Corporation. nanohub. omputational resoures have been used in this work. Referenes [] D. Mamaluy et al., Phys. Rev. B 7, (25). [2] D. Mamaluy et al., Semiond. Si. Teh. 9, 8 (24). [3] H. Khan et al., J. Comput. Eletron 6, 77 (27). [4] M. Luisier et al., Phys. Rev. B. 74, (26). [5] G. Klimek et al., IEEE Trans. Eletron Devies 54, 29 (27). [6] G. Klimek et al., VLSI Design 8, 79 (998). [7] P. Vogl et al., J. Phys. Chem. Solids 44, 365 (983). [8] S. Datta, Superlatt and Mirostrut. 28, 253 (2). [9] C. Rivas et al., Phys. Stat. Sol. (b) 239, 94 (23). [] R. Lake et al., J. Appl. Phys. 8, 7845 (996). [] S. Cauley et al., J. Appl. Phys., 2375 (27). [2] Y. X. Liu et al., Phys. Rev. B 54, 5675 (996). [3] E. Lind et al., Phys. Rev. B 68, 3332 (23). [4] R. Rahman et al., Phys. Rev. Lett. 99, 3643 (27). [5] T. B. Boykin et al., Phys. Rev. B 7, 52 (24).