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A SUPPLEMENTAL MATERIAL

# Non-Abelian Su(2) Lattice Gauge Theories in Superconducting Circuits

## Abstract

We propose a digital quantum simulator of non-Abelian pure-gauge models with a superconducting circuit setup. Within the framework of quantum link models, we build a minimal instance of a pure gauge theory, using triangular plaquettes involving geometric frustration. This realization is the least demanding, in terms of quantum simulation resources, of a non-Abelian gauge dynamics. We present two superconducting architectures that can host the quantum simulation, estimating the requirements needed to run possible experiments. The proposal establishes a path to the experimental simulation of non-Abelian physics with solid-state quantum platforms.

###### pacs:
03.67.Ac, 03.67.Lx, 85.25.Cp,11.15.Ha

Gauge invariance is a central concept in modern physics, being at the core of the standard model of elementary particle physics. In particular, invariances with respect to and gauge symmetries characterize the weak interaction and quantum chromodynamics (1); (2). In this sense, gauge theories represent a cornerstone in our understanding of the physical world and lie at the heart of diverse phenomena, such as the quark-gluon plasma or quantum spin liquids. In condensed matter physics, gauge fields can also emerge dynamically in relation to exotic many-body phenomena, like quantum Hall systems, frustrated magnets, or superconductors (3); (4); (5).

Lattice gauge theories (LGT) are non-perturbative discrete formulations that contribute to the analysis of key features of these models, such as color confinement or chiral symmetry breaking. Starting from the seminal work by Wilson in 1974 (6); (7), LGT have attracted a significant attention across several branches of theoretical physics. In the last decades (8), quantum Montecarlo simulations have achieved unprecedented accuracies in determining the whole hadronic spectrum of the standard model. However, understanding its full phase diagram from first principles, or simulating dynamical processes, remain out of reach of current numerical computations.

Quantum simulators (9) provide a new approach to solve complex long-standing problems in quantum physics. In a quantum simulator, a proper encoding of LGT can allow for the retrieval of information about ground state and dynamics, in a wide range of regimes. Previous works have considered Abelian (10); (11); (12); (13); (14); (15); (16); (17); (18); (19); (20); (21); (22) and non-Abelian LGT in optical lattices (25); (23); (24) (see also (26); (27) and references therein), both as analog and digital simulations (28). In these implementations, matter-gauge interactions are modeled as a second-order process from Hubbard-like interactions, while the simulation of pure-gauge dynamics remains more demanding.

In the last years, superconducting circuits have proven to be reliable devices that can host quantum information and simulation processes (29). The possibility to perform quantum gates with high fidelities, together with high coherence times, makes them ideal devices for the realization of digital quantum simulations (30); (33); (31); (32); (34), previously considered in ion-trap systems (35); (36); (37).

In this Letter, we propose a digital quantum simulation of a non-Abelian dynamical gauge theory in a superconducting device. We start by building a minimal setup, based on a triangular lattice, that can encode pure-gauge dynamics. The degrees of freedom of a single triangular plaquette of this lattice are encoded into qubits. We propose two implementations of this quantum simulator, using two different superconducting circuit architectures, as depicted in Fig. 1. We consider a setup in which six tunable-coupling transmon qubits are coupled to a single microwave resonator, and a device where six capacitively-coupled Xmon qubits stand on a triangular geometry, coupled to a central auxiliary one. We compute experimental requirements necessary to perform the simulation on one plaquette, and provide arguments for scaling to large lattices.

Lattice gauge theories.- LGT are discretized versions of a gauge theory. In a conventional approach of lattice gauge theories, space-time is discretized while ensuring a covariant formulation of the theory. In quantum simulations, there is no direct access to the time direction, that is fixed and continuous. Hence, the equivalent Hamiltonian formulation of lattice models is used. In this case, the space is discretized and the action of a local gauge invariant Hamiltonian characterizes a continuous dynamical evolution. This fictitious asymmetry of the time direction forces us to define a physical Hilbert space or Gauss law of the model. We make use of a set of discretized space points , and operators associated to each link, connecting two adjacent sites . A single operator has two implicit color indices , , that connect the -th component of the fermionic field at position with the -th component of the field at position .

In order to construct a non-Abelian theory built out of these operators, one has to build a Hamiltonian that is invariant under a generic gauge transformation

 u(→x,^μ)→eiθa(→x)τau(→x,^μ)e−iθa(→x+^μ)τa, (1)

where are the generators of the algebra in the corresponding representation, and the relative phase angles. It is straightforward to verify that the following Hamiltonian on a square lattice is gauge invariant

 H=J∑→xTr[u(→x,^μ)u(→x+^μ,^ν)u†(→x+^ν,^μ)u†(→x,^ν)]+H.c., (2)

where the trace is performed over the color indices, has energy units, and the directions and span the two-dimensional lattice. This Hamiltonian is a pure gauge operator, as it does not involve any fermionic operator and it is the minimum instance of a Wilson loop around a plaquette (1); (2).

We focus now on the construction of a non-Abelian quantum link model that mimics this gauge-invariant behavior in a finite-dimensional Hilbert space, suitable for quantum simulations. Notice that the framework of the quantum link models is valid for any compact Lie group (38). We use a four-dimensional Hilbert space or, equivalently, two qubits associated with each link. One can define link operators acting on this four-dimensional Hilbert space with the associated gauge generators that satisfy the algebra

 [Ga(→x),Gb(→y)]=iδ→x→y∑cϵabcGc(→x). (3)

To define the set of operators, we use a quantum link formulation (38), where the Hilbert space of a link is given by a set of bosonic modes that implement the Schwinger representation of the algebra, with , acting on two different sites on the link between the two adjacent vertices and . In this space, we build two sets of right and left generators , , on this finite-dimensional Hilbert space, using the bosonic modes  (39). We define the gauge generators as , where the sum over is taken among all the links of the lattice converging to a single vertex .

The representations of are quasi-real, therefore, ordinary and their dual representation (i.e., particle and antiparticle) are equivalent. Therefore, there are two multiplets with well-defined transformations (39). One can finally define the link operators, acting on the finite-dimensional Hilbert space of one link, in terms of Schwinger bosons on a given link (47); (48); (49),

 Uαβ=⎛⎝c↑L−ic†↓Lc↓Lic†↑L⎞⎠⋅(c†↑Rc†↓Ric↓R−ic↑R). (4)

These operators have the following commutation rules with the left and right operators

 [U(→y,^ν),Ra(→x,^μ)]=−U(→y,^μ)σa2δ→x→yδ^ν^μ,[U(→y,^ν),La(→x,^μ)]=σa2U(→y,^μ)δ→x→yδ^ν^μ, (5)

where we have defined the usual Pauli matrices , see also (39), while is defined as the identity operator. The commutation rules for a general gauge transformation follow in a straightforward way,

 ∏→ye−iθa(→y)Ga(→y) U(→x,^μ)∏→zeiθa(→z)Ga(→z) =eiθa(→x)σa2U(→x,^μ)e−iθa(→x+^μ)σa2, (6)

where the products over and are extended over the whole lattice. This equation shows that quantum link models are formulations of gauge-invariant models. In fact, one can notice how the action of a gauge transformation in Eq. (6) mimics Eq. (1).

A generic state in the local Hilbert space of a quantum link is given by the occupation of the operators . Since the total occupation per link, , is a constant of motion [see Eq. (4)], we restrict to the subspace with one occupied mode per link.

Notice that matter-gauge interactions in dimensions can be derived (39) in a second-order perturbation theory, considering extensions of previous works (50); (51). In the following, we will focus instead on pure-gauge two-dimensional interactions.

Triangular lattice.- The continuum limit of quantum link models and their thermodynamical properties are topics under active research. In related condensed matter models, the importance of the lattice geometry has been shown in fundamental aspect of the phase diagram, as the existence of confinement and de-confinement phases (52); (53). Quantum simulations on large lattices may provide insights on the continuum limit of LGT, and its dependence on the geometry of the underlying lattice. In this spirit, we consider a minimal implementation of a pure invariant model in a triangular lattice, by using triangular plaquettes, as depicted in Fig. 1c. In this case, the pure-gauge Hamiltonian on a single plaquette reads

 HT=−J Tr[U(→x,^μ)U(→x+^μ,^ν)U(→x+^μ+^ν,−^μ−^ν)]. (7)

This interaction corresponds to the magnetic term of a gauge invariant dynamics, which acts on closed loops. We focus on the magnetic term, since the representation of the electric counterpart is trivial.

Since this Hamiltonian commutes with gauge generators, , these are constant of motion, and one can define different gauge sectors with the initial values of the generators. Due to the triangular geometry and having one occupied mode per link, there are no global states that have zero eigenvalue for the gauge generators at every vertex in a single plaquette. In general, the absence of the zero-eigenvalue sector depends on the topology of the lattice, and is avoided, for example, in the periodic lattice of Fig. 2d. The different gauge sectors can be characterized by the eigenvalues of the operator , as in Fig. 2a,  2b, and 2c, see the supplemental material for additional details (39).

The local Hilbert space of a link is four dimensional, and it can be faithfully spanned by two qubits, called “position” and “spin” qubit . In this subspace, it is useful to define the operators , , such that the total Hamiltonian is written as  (39)

 HT= −J{Γ012Γ023Γ031+∑abcϵabcΓa12Γb23Γc31 (8) −∑a[Γ012Γa23Γa31+Γa12Γ023Γa31+Γa12Γa23Γ031]}.

Superconducting circuit implementation.- In order to simulate the interaction of Eq. (8), one can decompose its dynamics in terms of many-body monomials, and implement them sequentially with a digitized approximation (54). In a digital approach, one decomposes the dynamics of a Hamiltonian by implementing its components stepwise, (here and in the following ), for a total of  gates, with an approximation error that goes to zero as the number of repetitions grows. In a practical experiment, each quantum gate will be affected by a given error . By piling up sequences of such gates, for small gate errors , the total protocol will be affected by a global error, which is approximately the sum . This model for error accumulation has been proved in recent experiments (33); (32).

The effects of digitization in the simulation of LGT can be observed in Fig. 3. We have numerically integrated a Schrödinger equation regulated by the Hamiltonian in Eq. (8), choosing as initial state one of the states shown in Fig. 2a, for different simulated phases  (39). We compute both the evolution with the ideal Hamiltonian and with a digital sequence, in which we act with a single monomial at a time, repeating the protocol times. As a figure of merit, we compute the relative deviation from the ideal value of the gauge invariant, where stand for average values computed on the ideal (digitally simulated) state. The deviation goes to zero for large and small phases, defining surfaces at given error tolerance in the - space.

To simulate the pure-gauge interaction in a single triangular plaquette, we first consider a setup in which six tunable-coupling transmon qubits are coupled to a single microwave resonator (55); (56). Each tunable-coupling qubit is built using three superconducting islands, connected by two SQUID loops. Acting on these loops with magnetic fluxes, one can modify the coupling of the qubits with the resonator, without changing their transition frequencies. For further clarifications of the experimental setup involving tunable-coupling transmon qubits, see the Supplemental Material (39). By threading with magnetic fluxes at high frequencies, one can drive simultaneous red and blue detuned sidebands, and perform collective gates (57). Each many-body operator can be realized as a sequence of collective and single-qubit gates,

 UN=ei(π/4)∑i

where the indices and run from to . In this way, one can implement a generic (up to local rotations) -body evolution operator , with , using collective gates, and a number of single qubit gates which is upper bounded by , counting the single qubit gate in Eq. (9) and the rotations necessary to map to any -body operator. The simulation for one digital step will amount to implementing collective gates and a number of single qubit gates which is upper bounded by  (39).

We consider now the architecture of Fig. 1b, in which six Xmon qubits in a triangular geometry are capacitively coupled with an additional central ancillary qubit (58). In this case, the collective interactions can be decomposed and performed with pairwise -phase gates, using the central ancillary qubit to mediate non-nearest interactions. In this way, the quantum simulation of one digital step of the Hamiltonian in Eq. (8) will amount to realize  -Phase gates and a number of single-qubit rotations which is upper bounded by . For additional details on the experimental setup and gate counts, see (39).

To estimate the effects of experimental imprefections on the simulations, we plot in Fig. 4 the dynamics of a single plaquette, starting from a state. The observed periodic dynamical oscillations between the two paired states, depicted in Fig. 2b, are signaled by the periodic behavior of the overlap . The simulation is run for and steps, necessary to observe one flip between the two states and one full oscillation, respectively. On top of these oscillations, we plot fidelity bands that estimate the fidelity cap due to accumulated gate errors in the digital protocol, in the case of the two presented setups in Fig. 1, with collective and two-qubit gates. In order to see coherent oscillations, the fidelity of the two-body gates has to be one order of magnitude better with respect to the collective ones, with an error window of approximately in the collective gate case and a for the -phase gate setup. We assume a factor for the error loss on a single qubit gate. From the plots, it can be seen that, while the digital fidelity increases, the fidelity bands due to experimental errors broaden, affecting the simulation. The intersection of the digital fidelity line with the bands marks a regime dominated by experimental imperfections, for small , and one in which the error of the digital expansion prevails, for large .

The proposed simulation can be extended to larger lattices, where one could analyze dynamical properties of the confinement-deconfinement phase transition, which can be traced back to the breaking of the global center symmetry (59) in groups with a non-trivial center group, like the case. Moreover, such a LGT simulator can be used to perform quench experiments in a ladder configuration, analyzing breaking of gluon strings between pairs of particles and anti-particles (60); (61); (62).

In order to scale the quantum simulation to large qubit lattices, one has to consider that the accumulated gate error does not depend on the size of the lattice (33); (32), and only on the number of gates. The quantum resources necessary to run the simulation scale polynomially in the lattice size, and sub-polynomially in the digital error (63), making the whole protocol efficient. With a current total number of gates that exceeds 1000 (32); (34), we believe that in the near future simulations of LGT on large scales will be feasible. Furthermore, the possibility of performing quantum error correction for digital quantum simulations may drastically increase its effectiveness (64).

AM and LL acknowledge useful discussions with Rami Barends. ER thanks D. Banerjee, T. Calarco, M. Dalmonte, S. Montangero, U.-J. Wiese, P. Zoller for discussions, in the framework of quantum technologies for lattice gauge theories. We acknowledge financial support from Basque Government Grants IT472-10 and IT559-10, Spanish MINECO FIS2012-36673-C03-02, Ramón y Cajal Grant RYC-2012-11391, UPV/EHU Project No. EHUA14/04, UPV/EHU UFI 11/55, PROMISCE and SCALEQIT European projects.

## Appendix A Supplemental Material

In this supplemental material, we perform additional analysis that support the results obtained in the main article.

### a.1 Explicit Su(2) gauge operators in terms of Schwinger modes

In this section, we explicitly derive the link operators in terms of Schwinger bosons. Following (65), we define the operators , as right and left generators, acting on the finite-dimensional Hilbert space of the link,

 Ra(→x,^μ)=∑{α,β}∈{↑,↓}c†αR(→x,^μ)σaαβ2cβR(→x,^μ),La(→x,^μ)=∑{α,β}∈{↑,↓}c†αL(→x,^μ)σaαβ2cβL(→x,^μ), (10)

where are bosons that implement the Schwinger representation of the algebra, and the Pauli matrices

 σ(0)=(1001); σ(1)=(0110); σ(2)=(0−ii0); σ(3)=(100−1). (11)

The bosonic operators act on two different sites on the link between the two adjacent sites and and obey the commutation rules

 [Ra,Rb]=i∑cϵabcRc,  [Ra,Lb]=0,  [La,Lb]=i∑cϵabcLc, (12)

taking into account that they commute on different links (i.e., different , ). With the Schwinger representation, there are two multiplets with well-defined commutation relations

 [cαL,La]=∑γσaαγ2cγL,⎡⎣⎛⎝∑βσyαβc†βL⎞⎠,La⎤⎦=∑γσaαγ2⎛⎝∑βσyγβc†βL⎞⎠,[c†αR,Ra]=−∑γc†γRσaγα2,⎡⎣⎛⎝∑βcβRσyβα⎞⎠,Ra⎤⎦=−∑γ⎛⎝∑βcβLσyβγ⎞⎠σaγα2. (13)

Given these two multiplets, we can define the link operator as

 Uαβ=cαLc†βR+∑μνσyαμc†μLcνRσyνβ. (14)

We stress that these quantum link operators, or gauge fields, are associated with links between lattice points, transporting color information between the two points they connect. The two indices of the link operator are identified with the two ends of the link . The left (right) index is associated with the beginning (end) of the link as shown in Fig. 5. The gauge transformation acts on according to

 U(→x,^μ)=ei∑aθa(→x)σa2U(→x,^μ)e−i∑aθa(→x+^μ)σa2. (15)

Since in general and are different, our gauge-invariant equations will require invariance under right multiplication and left multiplication separately. An arbitrary gauge transformation can be built from individual gauge transformations at the point of the lattice. Therefore, it is sufficient to consider just a gauge transformation at the lattice site . Every link emanating from the point is affected by the gauge transformation generated by

 Ga(→x)=∑^μLa(→x−^μ,^μ)+Ra(→x,^μ), (16)

and the total color carried by a link is just the difference: .

The sum over non-Abelian electric fields emanating from a single point of the lattice is the lattice analog of the electromagnetic divergence of the electric field, . However, because the electric field itself varies from to along the link, there is an additional contribution given by the total color carried by the link. Thus, the generator is recast into: . The gauge invariance of the physical sector is defined by , i.e., at every vertex the state transforms locally as a singlet of . If we identify the local color density , then is the usual Gauss’ law. These are properties of non-Abelian lattice gauge theories, independent of their representation via quantum link models.

### a.2 Local Hilbert space of Su(2) quantum links

The local degrees of freedom in a link are encoded in the quantum state of four different bosonic modes . Different representations of this local Hilbert space are given by the occupation of these modes. The Hilbert space with one excitation in total is four-dimensional, and it can be decomposed in a spin degree of freedom with the symmetry group and a position degree of freedom.

A gauge invariant interaction commutes with the local generator of the at any vertex. As a consequence, the total angular momentum at any vertex of the lattice is a conserved quantity of the dynamics. In the simplest example of two quantum links, i.e. just one vertex, there are only two possible gauge sectors, the singlet sector where the total angular momentum around the vertex is zero, and the triplet sector where the total angular momentum is equal to one.

### a.3 Reduction to qubits - Triangular lattice

In the main text, we have considered the local four-dimensional Hilbert space spanned by the occupation of four different modes . By choosing the subspace with one excitation per link, we can faithfully represent its quantum state with two qubits, the position and spin qubits, . The excitations of these qubits are related to the left and right modes through and .

With these definitions, the left and right operators defined in the main text are recast into

 Unknown environment '% (17)

with , . The Hamiltonian for one plaquette can then be written down as

 HT=−JTr[U(→x,^μ)U(→x+^μ,^ν)U(→x+^μ+^ν,−^μ−^ν)]. (18)

In order to identify all the terms of the trace one can use the condition . The condition is met if and only if

 α=β=γ=0, α≠β≠γ≠0, α=0;β=γ≠0, β=0;α=γ≠0, γ=0;β=α≠0. (19)

The first condition gives term, the second terms, and terms for the last three equations, for a total of terms

 Tr[U12U23U31]=14{Γ012Γ023Γ031+∑abcϵabcΓa12Γb23Γc31−∑a[Γ012Γa23Γa31+Γa12Γ023Γa31+Γa12Γa23Γ031]}, (20)

where we have used the subscripts to represent the links , respectively. The pure-gauge Hamiltonian can then be written in terms of matrices as

 H=−J2{Γ012Γ023Γ031+∑abcϵabcΓa12Γb23Γc31−∑a[Γ012Γa23Γa31+Γa12Γ023Γa31+Γa12Γa23Γ031]}. (21)

Then, one can explicitly write this Hamiltonian in terms of up to six-body interactions between qubits, for instance

 Γ012Γ023Γ031=σxpos12σxpos23σxpos31Γa12Γb23Γc31=σypos12σypos23σypos31σam12σbm23σcm31∑aΓ012Γa23Γa31=σxpos12σypos23σypos31∑aσam23σam31. (22)

### a.4 Experimental setups

In this section we provide a brief overview of the experimental setups considered for the implementation of the proposed simulator. The first setup considered in the main text is comprised of six transmon qubits provided with tunable coupling to a single microwave resonator. A transmon qubit is a special instance of a Cooper pair box (66), operating at specific regimes.

A Cooper pair box is made of a piece of a superconducting island connected to a voltage source through a capacitor, and grounded via a Josephson junction. The temperature of these superconducting systems is low enough so that the quantum state of the setup can be defined by the number of Cooper pairs on the superconducting island. In order to add tunability and control to this quantum system, the single Josephson junction of the Cooper pair box is substituted with a SQUID loop, threaed with magnetic fluxes. These superconducting systems can be coupled to superconducting microwave resonators, which can be used to induce, via electromagnetic interactions, tunnelling of Cooper pairs in and out the superconducting island, as well as reading out the state of the system (67); (68). The Hamiltonian of the setup comprising a Cooper pair box coupled to a microwave resonator reads

 HS=4EC(n−ng)2−EJcosϕ+ωra†a+2βeVrmsn(a+a†). (23)

Here we have defined the quantized charge on the superconducting island, which is proportional to the number of Cooper pairs, the offset charge and the quantized flux that threads the SQUID loop. The () are bosonic operators that act upon the resonator field, of frequency . stands for the charging energy of the superconducting island, while is the Josephson energy of the dc-SQUID loop. The coupling coefficient is a function of the circuit capacitances, is the root mean square voltage of the resonator, and is the electron charge.

One of the main problems that arise in using Cooper pair boxes as controlled quantum systems is the high sensitivity of their quantum state to noise on the offset charge . By using large capacitive pads shunting the superconducting island, one can achieve insensitivity with respect to this noise. These Cooper pair boxes provided with large capacitive pads are known as transmon qubits (69). In this parameter regime, the Hamiltonian of Eq. (23) models a transmon qubit. It can be rewritten in a diagonal basis for the part which is not coupled to the resonator,

 HS→2∑i=0ωi|i⟩⟨i|+ωra†a+2∑i=0gi,i+1(|i⟩⟨i+1|+|i+1⟩⟨i|)(a+a†), (24)

where we have included the first three levels of this device. Higher levels are usually excluded from the quantum dynamics, as being more energetic and hard to access. The first three levels can be approximately modeled as the first three levels of an anharmonic oscillator, with a relative anharmonicity factor of . If is big enough, one can consider the transmon qubit device a two-level system (69).

The setups envisioned in this work use two variants of the transmon design. Each of the six devices sketched in Fig. 1a in the main text is made of three superconducting islands, connected via two SQUID loops, marked with different colors in the figure. The magnetic fluxes acting on these loops modify the quantum level structure of the system, and an appropriate manipulation of these fluxes in time allows for a fast modulation of the effective coupling to the resonator (see Eq. 24). This modulation does not affect the bare energy levels of the device (70); (71) and gives the possibility of performing red and blue sidebands upon the coupled qubit-resonator system, which, in turns, allows to generate many-body interactions (72).

The setup considered in Fig. 1b in the main text describes transmon qubits with improved design, the so called Xmon qubits (73). In this case, the shape of the capacitance pads is designed in order to improve coherence times of the device, by reducing coupling to material defects and radiative losses. The interactions among these qubits are mediated by direct capacitive couplings, not making use of a common superconducting resonator. For this reason, the many-body interactions are reduced in this case to nearest-neighbor two-qubit gates.

### a.5 Many-body interactions and gauge invariance for different sectors

In order to simulate the Hamiltonian associated with the non-Abelian model in Eq. (21), one can simulate stepwise the single many-body monomials that compose the total interaction. Each of them has the form of an -qubit string term

 HN=σi1σj2⋯σkN, (25)

where the set of superindices can assume values . Up to a maximum of local rotations, this general interaction can be mapped back to, e.g., . In what follows, we will show how to obtain this effective interaction in two different scenarios. In one case, one can use a set of tunable-coupling transmon qubits coupled to a single microwave resonator. In the second scenario, a set of charge-like qubits, e.g., transmon or Xmon qubits, are coupled to an additional ancillary qubit, for a total of qubits.

We consider the sequence of collective and single qubit gates (74); (72), as

 US(t)=ei(π/4)∑i

and obtain the set of interactions

 US(t)= e−igtσz1σx2⋯σxN,N=4n−1, US(t)= eigtσz1σx2⋯σxN,N=4n+1, US(t)= e−igtσy1σx2⋯σxN,N=4n, US(t)= eigtσy1σx2⋯σxN,N=4n−2. (27)

The collective interaction in Eq. (26) can be realized within a set of three-island transmon qubits, by threading the two SQUID loops embedded in each device with fast-tunable magnetic fluxes, tuned at appropriate frequencies (72). The general -body term of Eq. (25) can be simulated with this approach with a total of collective gates, and a number of single qubit gates which is upper bounded by . The Hamiltonian in Eq. (21) is composed of monomials, three-body term, six-body terms, and five-body terms. Therefore, the simulation for one digital step will amount to implement collective gates and a number of single-qubit gates which is bounded by .

One can implement the interaction in a set of capacitively-coupled charge-like qubits, by decomposing it into a set of -phase and single-qubit gates. First, let us consider the sequence of gates in Eq. (26) between qubits and an additional ancillary one, labeled by . We consider also that the gates are rotated about the axis of an angle . The sequence of gates in Eq. (26) can then be rewritten in the following way

 US,A(t)=ei(π/4)σzA∑Nj=1σzje−igtσxAe−i(π/4)σzA∑Nj=1σzj. (28)

This sequence of gates is equivalent to the following interactions

 US,A(t)= eigtσxAσz1⋯σzN,N=4n−1, US,A(t)= e−igtσxAσz1⋯σzN,N=4n+1, US,A(t)= e−igtσyAσz1⋯σzN,N=4n, US,A(t)= eigtσyAσz1⋯σzN,N=4n−2. (29)

If the ancillary qubit is initialized in an eigenstate of (), (), then the effective dynamics on the -qubit system would be equivalent to the interaction . The interaction can be realized as a sequence of gates between the ancillary qubit and all the other ones, . Each of these two-body gates can be realized as a sequence of two local -rotations and a -phase gate between the qubits and , . In fact, defining

 eiϕ/2e−i(ϕ/2)σz=(100eiϕ),CP1i(ϕ)=⎛⎜ ⎜ ⎜⎝100001000010000e−i2ϕ⎞⎟ ⎟ ⎟⎠, (30)

one has that

 e−i(ϕ/2)σz1e−i(ϕ/2)σziCP1i(ϕ)=ei(ϕ/2)σz1σzi. (31)

In this way, an arbitrary -body term can be encoded, up to a number of local rotations bounded by , with a sequence of -Phase gates and a number of single-qubit gates bounded by . Thus the quantum simulation of one digital step of the Hamiltonian in Eq. (21) will amount to realize -Phase gates and a number of single-qubit rotations which is bounded by .

The -Phase gate between two qubits can be performed by adiabatically changing the frequency of one qubit, bringing the level close to the anharmonic excitated level , to induce state-dependent phase accumulations. Unwanted errors due to leakage to the state are exponentially suppressed in the gate duration. For finite gate times errors can be minimized upon optimization in the Fourier spectrum of the control signal (75). This optimization has been carried out for gates in (76). For a generic quantum simulation involving -Phase gates, one should define the set of angles to be used in the experiment, and optimize over the duration and Fourier spectrum of the gate, to avoid unwanted leakage. This has been done in (77) by increasing the gate time from ns to ns, not observing significant state leakages up to phases of rads. Additional errors on the fidelity of the gate come from standard decoherence of the qubits, and control errors, see Ref. (76) for a discussion of the error budget.

Finally, to complete the information given in the main text, we show in this section, in Fig. 6, additional plots that complement Fig. 3 in the main text. The simulated gauge invariance is shown to converge for the singlet and triplet configurations on a single triangular plaquette.

The proposed experiment can be extended to larger simulated lattices, which are implementable by considering setups of superconducting qubits with enhanced connectivity. The required connectivity can be achieved by using several microwave resonators, acting as non-local quantum buses, or by considering superconducting pads that can provide with multiple capacitive couplings, as schematized in Fig. 7. For instance, we can envision, as shown in the main text, a ladder architecture that can support a singlet gauge invariant subspace, where non-Abelian string dynamics could be studied.

### a.6 Scaling of errors for larger lattices

It has been shown (78) that the scaling of the number of steps needed to achieve a digital error in a Suzuki-Lie-Trotter decomposition is almost linear in the simulated phase of the dynamics and sub-polinomial in , according to the formula

 Ns≤2m52k(m||H||t)1+1/2kϵ1/2k. (32)

In the formula is the number of terms that compose a generic Hamiltonian , and is the fractal degree of approximation (79), which we set to . For one plaquette one has , three-body term, six-body terms, and five-body terms, all present in the system with the energy . Considering that the norm of an arbitrary product of Pauli matrices is , , one has, using a triangular inequality, that the norm of the Hamiltonian for one plaquette is bounded by . As a consequence, the number of steps necessary to implement a dynamics with error for the time on a lattice of plaquettes is bounded by

 Ns≤2300N(N|J|t)3/2√ϵ. (33)

Notice that this formula gives a very loose bound which is not dependent on the underlying structure of the Hamiltonian. Nevertheless, it shows that the scaling of the number of steps is efficiently solvable by a quantum simulator, not being exponential in the resources used. The exact scaling of the number of steps cannot be known (up to an exact solution of the problem, which is not addressable numerically for larger lattices), however it is useful to extract the dependence from the upper-bound formula in Eq. (33). Assuming independent experimental errors on the single Trotter steps, one can relate the total number of steps to the required experimental fidelity for the implementation of a single Hamiltonian term. We impose an accumulated experimental error of the same order of magnitude of the Trotter error . This enforces the error on the single step to be of the order . Besides these issues on error scaling and accumulation, the accesible simulation time has to be smaller than and of the single qubits, defining a system of necessary conditions for the simulation of a lattice of plaquettes

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩Ns≤2300N(N|J|t)3/2√ϵϵs∼ϵNst≪T1,T2. (34)

### a.7 Matter-field Su(2) Gauge Interactions

In the main text we have focused on pure gauge dynamics with plaquette interactions, where usual perturbative approaches do not get energy scales which are big enough. In this section, we point out the basic elements needed for the simulation of the simplest gauge model coupled to a matter field in -dimensions. The basic ingredients and the validity of the different energy-scale regimes have been already discussed in the simulation of Abelian gauge models in trapped ions and superconducting circuits (80); (81). This section can be seen as a generalization to non-Abelian setups.

The ideal interaction Hamiltonian that we would like to simulate is given by

 Hint=∑ib†(i)U(i,i+1)b(i+1)+H.c.=∑iαβb†α(i){[cαL(i)c†βR(i+1)]+∑μν[σyαμc†μL(i)cνR(i+1)σyνβ]}bβ(i+1)+H.c., (35)

where and are hard-core bosonic modes, with , and .

In terms of these modes, the former Hamiltonian has two local (gauge) symmetries:

• gauge invariance around every site with generators:


text:
## Stream: Is there code for X?

### Topic: le_or_le

#### Johan Commelin (Mar 04 2021 at 05:53):

Does it make sense to add this to mathlib? (I couldn't find it.)
I often find myself wanting to case on this, and invoking wlog every time is somewhat slow.

lemma le_or_le {α : Type*} [linear_order α] (a b : α) :
a ≤ b ∨ b ≤ a :=
(le_or_lt a b).imp id le_of_lt

#### Johan Commelin (Mar 04 2021 at 06:01):

Aah, the wonders of a predictable naming convention :rofl:

#### Kevin Buzzard (Mar 04 2021 at 07:19):

lt_irrefl is another one which takes some getting used to -- I would always look for not_lt_self.

#### Mario Carneiro (Mar 04 2021 at 07:22):

I think we already have le_or_ge, so I don't think it would be a bad idea to have le_or_le as an alias for le_total

#### Patrick Massot (Mar 04 2021 at 07:50):

I agree we could have aliases for both le_total and lt_irrefl.

#### Yury G. Kudryashov (Mar 04 2021 at 07:55):

We also have an alias docs#has_lt.lt.false

Last updated: May 17 2021 at 16:26 UTC

format: plain


text:
# Books / Digital Text

## Appendix: Schumpeter and the Zero Rate of Interest

The late Professor Joseph Schumpeter pioneered a theory of interest which holds that the rate of interest will be zero in the evenly rotating economy. It should be clear from the above discussion why the rate of interest (the pure rate of interest in the ERE) could never be zero. It is determined by individual time preferences, which are all positive. To maintain his position, Schumpeter was forced to assert, as does Frank Knight, that capital maintains itself permanently in the ERE. If there is no problem of maintenance, then there appears to be no necessity for the payment of interest in order to maintain the capital structure. This view, treated above, is apparently derived from the static state of J.B. Clark and seems to follow purely by definition, since the value of capital is maintained by definition in the ERE. But this, of course, is no answer whatever; the important question is: How is this constancy maintained? And the only answer can be that it is maintained by the decisions of capitalists induced by a rate of interest return. If the rate of interest paid were zero, complete capital consumption would ensue.42

The conclusive Mises-Robbins critique of Schumpeter's theory of the zero rate of interest, which we have tried to present above, has been attacked by two of Schumpeter's disciples.43 First, they deny that constancy of capital is assumed by definition in Schumpeter's ERE; instead it is “deduced from the conditions of the system.” What are these conditions? There is, first, the absence of uncertainty concerning the future. This, indeed, would seem to be the condition for any ERE. But Clemence and Doody add: “Neither is there time preference unless we introduce it as a special assumption, in which case it may be either positive or negative as we prefer, and there is nothing further to discuss.” With such a view of time preference, there is indeed nothing to discuss. The whole basis for pure interest, requiring interest payments, is time preference, and if we casually assume that time preference is either nonexistent or has no discernible influence, then it follows very easily that the pure rate of interest is zero. The authors’ “proof” simply consists of ignoring the powerful, universal fact of time preference.44

• 42. See Mises, Human Action, pp. 527–29. Also see Lionel Robbins, “On a Certain Ambiguity in the Conception of Stationary Equilibrium” in Richard V. Clemence, ed., Readings in Economic Analysis (Cambridge: Addison-Wesley Press, 1950), I, 176 ff.
• 43. Richard V. Clemence and Francis S. Doody, The Schumpeterian System (Cambridge: Addison Wesley Press, 1950), pp. 28–30.
• 44. As has been the case with all theorists who have attempted to deny time preference, Clemence and Doody hastily brush consumers’ loans aside. As Frank A. Fetter pointed out years ago, only time preference can integrate interest on consumers’ as well as on producers’ loans into a single unified explanation. Consumers’ loans are clearly unrelated to “productivity” explanations of interest and are obviously due to time preference. Cf. Clemence and Doody, The Schumpeterian System, p. 29 n.


text:
# What is wrong with this problem

We know that:

$(a^n)^m=a^{nm}$

From this we have: $-3^3=[(-3)^2]^\frac{3}{2}=(3^2)^\frac{3}{2}=27$

Find what's wrong

• We do not know that when $m,n$ are not integers, unless $a$ is positive. Jan 23, 2013 at 15:29
• I'm surprised no one's mentioned $-3^3=-(3^3)$, not $(-3)^3$.
– Mike
Jan 23, 2013 at 16:06
• en.wikipedia.org/wiki/… Jan 23, 2013 at 16:43

$\exists m,n\in\Bbb Q$ such that $(a^n)^m \not= a^{nm}$ if $a<0$.

• In fact the symbol $\neq$ is not really the right one. It implies 'is not the same as', but one needs a symbol that implies something like 'is not necessarily the same as' but I guess that one doesn't exist. Jan 23, 2013 at 17:04
• I meant that the statement is not true for all values of m and n. Jan 23, 2013 at 17:06

The "law of exponents" that you cite:

$$\large (a^n)^m=a^{nm}$$

applies PROVIDED $\bf{a \gt 0}$.

Here, though, we have $\,\bf{a = -3 \lt 0}$, and hence:

$$-3^3=[(-3)^2]^{\large\frac{3}{2}}=(3^2)^{\large\frac{3}{2}}=27\quad \Longleftarrow \;\text{ False}.$$ $$-3^3 = -(3^3) = -[(3^2)^{\large \frac{3}{2}}] = -(9^{\large \frac{3}{2}}) = -27\quad \Longleftarrow \; \text{ True}$$ But why the trouble? It is a very straightforward computation: $$-3^3 = -(3^3) = -(3\cdot 3\cdot 3) = -27$$

You can not do that. I mean $(-3)^1\neq\sqrt{(3)^2}$, since $-3<0$.

• nice example...+1 Jan 23, 2013 at 15:36
• @amWhy: Thanks so much! Jan 23, 2013 at 15:40

when you are increasing the power of a by 3/2, you are actually cubing a and then taking square root, and you know that when to take the square root of a number, you have to consider both positive and negative roots. In this case you are neglecting the negative root which was your answer.

Remember one thing:: If you want to solve an equation and you are increasing the power of unknown(x) then you might get some extra values of x as solutions which may not be solution of your original equation.


text:
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# Nanometer-thick copper films with low resistivity grown on 2D material surfaces

## Abstract

Thin Copper (Cu) films (15 nm) are deposited on different 2D material surfaces through e-beam deposition. With the assist of van der Waals epitaxy growth mode on 2D material surfaces, preferential planar growth is observed for Cu films on both MoS2 and WSe2 surfaces at room temperature, which will induce a polycrystalline and continuous Cu film formation. Relative low resistivity values 6.07 (MoS2) and 6.66 (WSe2) μΩ-cm are observed for the thin Cu films. At higher growth temperature 200 °C, Cu diffusion into the MoS2 layers is observed while the non-sulfur 2D material WSe2 can prevent Cu diffusion at the same growth temperature. By further increasing the deposition rates, a record-low resistivity value 4.62 μΩ-cm for thin Cu films is observed for the sample grown on the WSe2 surface. The low resistivity values and the continuous Cu films suggest a good wettability of Cu films on 2D material surfaces. The thin body nature, the capability to prevent Cu diffusion and the unique van der Waals epitaxy growth mode of 2D materials will make non-sulfur 2D materials such as WSe2 a promising candidate to replace the liner/barrier stack in interconnects with reducing linewidths.

## Introduction

In silicon (Si) industry, copper (Cu) has been widely used for the application of interconnects due to its low resistivity1,2,3. On the other hand, the high diffusivity of Cu will result in Cu atom diffusion into the dielectric layers such that short circuit and therefore, system failure will be observed for Si chips4,5,6,7. Therefore, to avoid the problem of Cu diffusion, diffusion barrier layers between Cu wires and dielectrics are required for Cu interconnects. In nowadays, thin TaN layers (~ 3 nm) are commonly adopted as the diffusion barrier layer to isolate the Cu wires (~ 20 nm in widths/heights) from the surrounding dielectrics7,8. For a better adhesion between Cu and the TaN diffusion barrier layer, an additional liner layer Ta (~ 1 nm) is also required9. In this case, the Cu/Ta/TaN stack has become the standard interconnect structure in modern semiconductor industries. However, with the further shrinkage in the device line widths below 3 nm, interconnect scaling down to sub-20 nm will be required in the future. With the reduction of Cu wire thicknesses, its resistivity will increase remarkably due to the electron scattering by phonons, point defects, impurities and grain boundaries9,10,11,12,13,14. To avoid significant resistivity increasing with reduced Cu wire thickness, an alternate choice would be the thinner barrier/liner layers. However, when the TaN film thickness is reduced to below 3 nm, the barrier layer is no longer able to prevent the Cu diffusion to the surrounding dielectrics14. In this case, a new material with the capability of blocking Cu diffusion in a few nanometer thicknesses will become an urgent need for further interconnect scaling. Under this scenario, since mono-layer 2D materials are of few angstroms in thicknesses, they will become a promising candidate for this application. Besides the thin body nature of 2D materials, in previous publications, we have demonstrated vertical 2D material hetero-structures with epitaxially grown elemental 2D materials such as antimony (Sb), germanium (Ge) and tin (Sn) grown on molybdenum disulfide (MoS2) surfaces15,16. The results have revealed that the epi-layer formation through the van der Waals epitaxy growth mode on 2D material surfaces has less dependent to the substrate lattice structures. By further expanding this concept to metal crystals, van der Waals epitaxy on 2D material surfaces will help to improve the crystallinity of thick gold (Au) films grown on MoS2 surfaces17. The same mechanism will also help to improve the continuity of the nanometer-thick Au film such that close-to-theory sheet resistance will be obtained for 10 nm Au film grown on MoS2 surfaces18. The continuous thin Au films grown on MoS2 surfaces have also demonstrated good adhesion and wettability at the metal/2D material interfaces. Therefore, by using the thin 2D materials to replace the Ta/TaN stack, further interconnect scaling may be achieved. The unique van der Waals epitaxy growth mode on 2D material surfaces will be advantageous for the formation of continuous metal films with improved crystallinity such that even lower resistivity values can be obtained.

In this paper, we have deposited 15 nm Cu films on MoS2 and WSe2 surfaces by using the e-beam deposition system. With the assist of van der Waals epitaxy growth mode, polycrystalline Cu films are obtained on 2D material surfaces at room temperature (RT). By using the non-sulfur 2D material such as WSe2, Cu diffusion can be avoided at higher growth temperatures. By further increasing the deposition rates, a record-low resistivity value 4.62 μΩ-cm for thin Cu films is observed for the sample grown on the WSe2 surface. The thin thicknesses down to few atomic layers and good wettability of WSe2 to Cu have made it a promising candidate to replace the liner/barrier stack in interconnects with reducing linewidths.

## Results and discussions

### Thin copper films grown on different substrates

To investigate the influence of van der Waals epitaxy to the thin Cu films, 15 nm Cu were deposited on 300 nm SiO2/Si and tri-layer MoS2/c-plane sapphire substrates at RT by using a e-beam deposition system with 0.1 Å/s deposition rate. The 500 $$\times$$ 500 nm2 atomic force microscopy (AFM) images of the two samples are shown in Fig. 1a. As shown in the figure, Cu clusters with deep trenches are observed for the sample grown on the SiO2/Si substrate, which indicates that the dangling bonds on SiO2 surfaces may hinder the Cu atom migration and therefore, island formation instead of planar film growth will be observed on SiO2 surfaces. Different from the Cu film grown on SiO2 surfaces, a more continuous Cu film is observed on the MoS2 surface. Although Cu clusters are still observed for the sample, the shallower trenches between the clusters suggest that the van der Waals epitaxy growth mode on 2D material will help the planar growth of thin metal films. The root-mean-square (RMS) roughness values of the Cu films deposited on SiO2 and MoS2 surfaces are 1.67 and 1.04 nm, respectively, which is consistent with the observation that deeper trenches are observed for the Cu film deposited on the SiO2 surface. As a result, compared with the resistivity value 19.71 μΩ-cm for the sample grown on the SiO2/Si substrate, a much lower resistivity 6.07 μΩ-cm will be observed for the sample grown on the MoS2 surface, which is close to the ~ 5 μΩ-cm resistivity obtained from 15 nm Cu film grown on mechanically exfoliated MoS2 surfaces19. The 2θ − θ curves of the two samples measured by the X-ray diffraction system (XRD) are shown in Fig. 1b. Except for the substrate peaks, an additional Cu (111) peak is observed only for the Cu film grown on the MoS2 surface, which suggests that besides planar film growth, a better crystallinity is also obtained for the Cu film grown on MoS2 surfaces. The observation of Cu (111) peak alone may suggests that most of the Cu domains grown on MoS2 surfaces are well-aligned with (111) direction. Further investigation is required in the future to verify this point. On the other hand, it is expected that different orientations of the Cu domains are obtained on the SiO2 surface. Therefore, compared with the well-aligned (111) direction for the Cu domains on the MoS2 surface, the weakened XRD peaks of the Cu domains are not observed for the Cu film deposited on the SiO2 surface. The improved crystallinity of the Cu films grown on 2D material surfaces may help to reduce the inelastic electron scatterings on the Cu grain boundaries.

### Copper films grown on MoS2 surfaces at different temperatures

To further investigate the influence of growth temperatures to the morphologies and crystallinity of the thin Cu films, two more samples with 15 nm Cu deposited on MoS2 surfaces at 100 and 200 °C are prepared. The 1 × 1 $$\upmu$$m2 AFM images of the two samples are shown in Fig. 2a. The 1 × 1 $$\upmu$$m2 AFM image of the sample prepared at RT is also shown in the figure. With increasing growth temperatures, significant Cu grain coalescence is observed. Flattened plateau with deeper trenches in between is also observed at higher growth temperatures. The phenomenon of Cu grain coalescence will also result in an increasing RMS roughness values from 1.04, 1.44 to 2.23 nm for the Cu films grown at RT, 100 and 200 °C, respectively. The results may indicate improved crystallinity of the thin Cu films with increasing growth temperatures. The normalized Cu (111) XRD peaks of the three samples are shown in Fig. 2b. As shown in the figure, the full widths at the half maximum (FWHM) of the Cu (111) XRD peak decreases from 0.69, 0.64 to 0.61 degree for the three samples prepared at RT, 100 °C and 200 °C, respectively, which suggests that improved crystallinity/larger Cu grains are obtained with increasing growth temperatures. The results are consistent with the observation of enlarged Cu grains with increasing growth temperatures from the AFM images. However, although improved crystallinity is obtained at higher growth temperatures, the resistivity values of the thin Cu films will increase from 6.07 (RT), 8.16 (100 °C) to 40.2 (200 °C) μΩ-cm with increasing growth temperatures. The less continuous Cu films resulted from the Cu grain coalescence with increasing growth temperatures may be the main mechanism responsible for this phenomenon. Since the role of MoS2 is to replace the Ta/TaN stack in interconnects, its effect to block the Cu diffusion is a critical issue for this purpose. The cross-sectional high-resolution transmission electron microscope (HRTEM) images of the three samples are shown in Fig. 2c. As shown in the figure, clear tri-layer MoS2 with polycrystalline Cu films are observed for the two samples prepared at RT and 100 °C, respectively. However, for the sample prepared at 200 °C, the Cu/MoS2 interfaces are less clear. Some MoS2 layers seems to be lifted away from the sapphire substrates, which may indicate Cu diffusion into the MoS2 films at 200 °C. To investigate this phenomenon, the high-angle annular dark field (HAADF) mappings of Cu, S and Mo elements for the sample grown at 200 °C are shown in Fig. 2d. The observation of Cu signals in the MoS2 layer suggests that Cu atoms will diffuse into MoS2 at 200 °C, which is also observed for the sample grown at 100 °C. The HAADF mappings of Cu, S and Mo elements for the sample grown at 100 °C are shown in the supplementary information Fig. S1. As shown in Fig. 2c, the diffused Cu atoms may lift some MoS2 layers away from the sapphire substrates. In this case, both Mo and S signals will also be observed in the Cu layer near the Cu/MoS2 interface.

### Copper films grown on WSe2 surfaces

Since Cu atoms will easily react with sulfur, it is possible that the defects with Mo vacancies will induce Cu diffusion into the MoS2 layers. The element sulfur is also a potential contaminant for semiconductor fabrication lines. In this case, a non-sulfur 2D material should be adopted for the application of liner/barrier layer in interconnects. To demonstrate this point, wafer-scale tri-layer tungsten diselenide (WSe2) films are grown on sapphire substrates by selenizing pre-deposited tungsten films at 950 °C. The picture and the cross-sectional HRTEM image of the standalone WSe2 sample showing its uniformity are shown in the supporting information (Fig. S2). By using the WSe2/sapphire samples as the new substrates, a 15 nm Cu film is deposited on the WSe2 surface at RT. The 2θ–θ curve of the sample measured by XRD is shown in Fig. 3a. The X-ray curve of the Cu film grown on the MoS2 surface at RT is also shown in the figure for comparison. Besides the substrate peak, the Cu (111) peak is observed on both MoS2 and WSe2 samples. The results indicate that compared with the Cu film grown on the SiO2 surface (Fig. 1), better crystallinity is obtained for the thin Cu films grown on both MoS2 and WSe2 surfaces. The same van der Waals epitaxy of the thin Cu film grown on the MoS2 surface may also take place on the WSe2 surface. The 1 × 1 $$\upmu$$m2 AFM image of the sample grown on the WSe2 surface is shown in Fig. 3b. Compared with the sample grown on the MoS2 surface (Fig. 2a), a similar morphology is observed for the sample grown on the WSe2 surface, which indicates that the unique van der Waals epitaxy growth mode will take place on different 2D material surfaces. The preferential planar film growth instead of island formation will improve the continuity of the thin Cu films deposited on both MoS2 and WSe2 surfaces. In this case, a similar resistivity value 6.66 μΩ-cm with the sample grown the MoS2 surface (6.07 μΩ-cm) will be observed for the 15 nm Cu film grown on the WSe2 surface.

To investigate the influence of growth temperatures to the Cu diffusion in the WSe2 samples, two more samples with 15 nm Cu deposited on WSe2 surfaces at 100 and 200 °C are prepared by using the e-beam deposition system with 0.1 Å/s deposition rate. Similar with the samples grown on MoS2 surfaces, Cu grain coalescence and decreasing FWHMs of the Cu (111) XRD peak are observed for the WSe2 samples with increasing growth temperatures (Fig. S3 in the supporting information). The results confirm again that the same van der Waals epitaxy occurs on both MoS2 and WSe2 surfaces. The cross-sectional HRTEM images of 15 nm Cu films grown on WSe2 surfaces at RT, 100 and 200 °C are shown in Fig. 4a. Unlike partial MoS2 liftoff from the sapphire substrates for the sample grown on the MoS2 surface at 200 °C, polycrystalline Cu films with clear tri-layer WSe2 are observed for all the three samples. The results suggest that WSe2 layers may effectively prevent Cu diffusion. The HAADF mappings of Cu and W elements for the sample grown at 200 °C are shown in Fig. 4b. A clear separation of Cu and Se signals are observed in the figure. The results indicate that by using non-sulfur 2D materials as the barrier layer, Cu diffusion can be effectively avoided. Compared with the ~ 4 nm thick Ta/TaN stacks, the thickness of tri-layer WSe2 is around 2.0 nm. Further thickness reduction is possible by reducing the layer numbers of the WSe2 films. The thinner line/barrier stacks of interconnects will leave a large volume for Cu films such that further size reduction is possible for the interconnects. The relatively low resistivity values 6.07 and 6.66 μΩ-cm and the continuous Cu films observed on MoS2 and WSe2 surfaces at RT suggest a good wettability of Cu films on 2D material surfaces. The thin body nature, the capability to prevent Cu diffusion and the unique van der Waals epitaxy growth mode of 2D materials will make WSe2 a promising candidate to replace the Ta/TaN stack in interconnects with reducing linewidths.

Although the van der Waals epitaxy growth mode on 2D material surfaces is advantageous for planar growth of metal films. The polycrystalline nature of grown 2D materials with wafer scale may hinder the adatom migration and induce local island formation on the 2D material surfaces. In this case, compared with exfoliated 2D material flakes, higher resistivity values will be observed on grown 2D material surfaces with the same metal film thicknesses. As discussed in the previous section, although a resistivity value 6.07 μΩ-cm can be achieved for the 15 nm Cu film grown on the MoS2 surface at RT, a lower resistivity value ~ 5 μΩ-cm is still observed on exfoliated MoS2 flakes with the same Cu thickness20. As we have observed for the samples with different growth temperatures, the Cu grain coalescence will improve the crystallinity of the film. The same mechanism will also induce a less continuous film such that higher resistivity values will be observed. Therefore, one possible approach to solve this problem is to increase the adatom density on the substrate surfaces such that a more continuous film can be obtained. 15 nm Cu films with deposition rates 0.5 and 1.0 Å/s are deposited on WSe2 surfaces by using the e-beam deposition system. The resistivity values obtained for the two samples are 5.22 and 4.62 μΩ-cm, respectively. The resistivity values of Cu films reported in previous publications are shown in Fig. 5. The resistivity value 1.68 μΩ-cm of Cu in nature is also shown in the figure. As shown in the figure, the resistivity value 4.62 μΩ-cm is the lowest value reported in literature. Given the fact that polycrystalline instead of single-crystal films are obtained for wafer-scale 2D materials at this stage, it is possible to achieve further reduction in the resistivity value for the thin Cu films if improved crystallinity can be achieved for grown 2D materials in the future. Nevertheless, the recorded low resistivity value of the thin Cu film grown on the WSe2 surface has already demonstrated its potential to replace the Ta/TaN stack in interconnects. Besides the direct growth of 2D materials on different substrates, the film transferring can be an easy approach to facilitate the integration of 2D materials with other substrates. After transferring the tri-layer WSe2 to a 300 nm SiO2/Si substrate, 15 nm Cu is deposited on the WSe2 surface by using the e-beam deposition system with the deposition rate 1.0 Å/s at RT. Compared with the value 4.62 μΩ-cm obtained for the sample grown on the un-transferred WSe2 surface, a similar resistivity value 5.43 μΩ-cm is observed for this sample. Despite the contamination and damages introduced during the transferring procedure, the results have demonstrated that van der Waals epitaxy is still the main mechanism of epi-layer growth on 2D material surfaces.

## Conclusion

We have demonstrated 15 nm Cu film deposited on different 2D material surfaces. With the assist of van der Waals epitaxy growth mode on 2D material surfaces, preferential planar growth is observed for Cu films on both MoS2 and WSe2 surfaces at room temperature, which will induce a polycrystalline and continuous Cu film formation. At higher growth temperature 200 °C, Cu diffusion into the MoS2 layers is observed while the non-sulfur 2D material WSe2 can prevent Cu diffusion at the same growth temperature. By further increasing the deposition rates, a record-low resistivity value 4.62 μΩ-cm for thin Cu films is observed for the sample grown on the WSe2 surface. After transferring the tri-layer WSe2 to a 300 nm SiO2/Si substrate, a similar resistivity value 5.43 μΩ-cm is obtained with 15 nm Cu deposited on the WSe2 surface. The low resistivity values of thin Cu films grown on transferred or un-transferred WSe2 surfaces have provided varieties for the 2D material integration with other materials in practical applications. These results have demonstrated that non-sulfur 2D materials such as WSe2 can be a promising candidate to replace the liner/barrier stack in interconnects with reducing linewidths.

## Methods

For the growth of tri-layer MoS2, the Mo metal was deposited on the sapphire substrates by using a radio-frequency (RF) sputtering system. We used 30 W for constant sputtering power and 5 × 10−3 Torr with a 30 sccm gas flow of argon (Ar) for background pressure. The deposition time is 37 s. for the preparations of MoS2 layer. After the metal deposition, the sample was placed at the center of the hot furnace for sulfurization. 200 sccm Ar gas was used as carrier gas while the pressure was kept at 50 Torr. The sulfurization temperature was kept at 850 °C with 0.25 g of sulfur (S) powder for 20 min. For the growth of tri-layer WSe2, a similar transition metal deposition and selenization procedure is adopted. The W metal film was deposited on the sapphire substrate by using the same RF sputtering system. Similar deposition conditions as the Mo deposition are adopted for the W deposition except for the higher sputtering power 40 W. The deposition time is 25 s. for the W deposition. After the metal deposition, the sample was placed at the center of the hot furnace for selenization. 85 sccm Ar gas was used as carrier gas while the pressure was kept at 100 Torr. The selenization temperature was kept at 950 °C with 0.2 g of Se powder for 30 min. The deposition rate and the thickness of the Cu films are measured by the quartz crystal microbalance (QCM) which is equipped with the e-beam system. The resistivity values were obtained through the equation $$\rho$$ = RS × (film thickness), where RS is the sheet resistance obtained through four-point measurements. For the four-point measurements, a four-pin probe station KSR-4 with BeCu probe tip and tip pitch 1 mm is adopted. Keithley 2400 SourceMeter is used to measure the sheet resistance of the samples. The cross-sectional HRTEM images measurement are obtained by using a JEOL JEM-2800F transmission electron microscopy system operated at 200 kV. The XRD measurement are done by using a Bruker New D8 Discover XRD system with Cu Kα radiation (λ = 1.5406 Å). The AFM measurement are done by using a BRUKER Dimension ICON AFM system.

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6. Wojcik, H. et al. Cu barrier properties of very thin Ta and TaN films. IEEE Int. Interconnect. Technol. Conf. 1, 167–170 (2014).

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8. Graham, R. L. et al. Resistivity dominated by surface scattering in sub-50 nm Cu wires. Appl. Phys. Lett. 96, 042116 (2010).

9. Plombon, J., Andideh, E., Dubin, V. M. & Maiz, J. Influence of phonon, geometry, impurity, and grain size on copper line resistivity. Appl. Phys. Lett. 89, 113124 (2006).

10. Lo, C.-L. et al. Opportunities and challenges of 2D materials in back-end-of-line interconnect scaling. J. Appl. Phys. 128, 080903 (2020).

11. Mayadas, A. & Shatzkes, M. Electrical-resistivity model for polycrystalline films: The case of arbitrary reflection at external surfaces. Phys. Rev. B 1, 1382 (1970).

12. Zhang, W. et al. Geometry effect on impurity incorporation and grain growth in narrow copper lines. J. Electrochem. Soc. 152, C832 (2005).

13. Chawla, J., Gstrein, F., O’Brien, K., Clarke, J. & Gall, D. Electron scattering at surfaces and grain boundaries in Cu thin films and wires. Phys. Rev. B 84, 235423 (2011).

14. Roberts, J. M.; Kaushik, A. P.; Clarke, J. S. Resistivity of sub-30 nm copper lines. 2015 IEEE International Interconnect Technology Conference and 2015 IEEE Materials for Advanced Metallization Conference, 341–344 (2015).

15. Lo, C.-L. et al. Studies of two-dimensional h-BN and MoS2 for potential diffusion barrier application in copper interconnect technology. NPJ 2D Mater. Appl. 1, 1–7 (2017).

16. Chen, H.-A. et al. Single-crystal antimonene films prepared by molecular beam epitaxy: selective growth and contact resistance reduction of the 2D material heterostructure. ACS Appl. Mater. Interfaces. 10, 15058–15064 (2018).

17. Chen, K.-C. et al. Multi-layer elemental 2D materials: antimonene, germanene and stanene grown directly on molybdenum disulfides. Semicond. Sci. Technol. 34, 105020 (2019).

18. Chen, K.-C. et al. Waals epitaxy of large-area and single-crystalline gold films on MoS2 for low-contact resistance 2D–3D interfaces. ACS Appl. Nano Mater. 3, 2997–3003 (2020).

19. Zhang, Y.-W., Wu, B.-Y., Chen, K.-C., Wu, C.-H. & Lin, S.-Y. Highly conductive nanometer-thick gold films grown on molybdenum disulfide surfaces for interconnect applications. Sci. Rep. 10, 1–6 (2020).

20. Shen, T. et al. MoS2 for enhanced electrical performance of ultrathin copper films. ACS Appl. Mater. Interfaces. 11, 28345–28351 (2019).

21. Lo, C. L. et al. Enhancing interconnect reliability and performance by converting tantalum to 2D layered tantalum sulfide at low temperature. Adv. Mater. 31, 1902397 (2019).

## Acknowledgements

This work was supported in part by the projects MOST 108-2221-E-001-017-MY3 and MOST 110-2622-8-002-014 funded by the Ministry of Science and Technology, Taiwan, and in part by the iMATE project AS-iMATE-109-41 funded by Academia Sinica, Taiwan. The authors would also like to thank Integrated Service Technology Inc. for their assistance in HRTEM measurements.

## Author information

Authors

### Contributions

Y.-W.L., D.-J.Z., P.-C.T. and C.-T.C. did the material growth/characterizations and electrical measurements of the samples. Y.-W.L., W.-C.T., and S.-Y.L. clarified the experimental data and came out with the manuscript. S.-Y.L. is the group leader and conducted the investigation of this work.

### Corresponding author

Correspondence to Shih-Yen Lin.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Liu, YW., Zhang, DJ., Tsai, PC. et al. Nanometer-thick copper films with low resistivity grown on 2D material surfaces. Sci Rep 12, 1823 (2022). https://doi.org/10.1038/s41598-022-05874-9

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• DOI: https://doi.org/10.1038/s41598-022-05874-9


text:
Or rather, what can the shapes of proofs tell us about them?

 April CACM source

Juris Hartmanis did much to lay the landscape of computational complexity beginning in the 1960s. His seminal paper with Richard Stearns, “On the Computational Complexity of Algorithms,” was published 50 years ago this month, as observed by Lance Fortnow in his blog with Bill Gasarch. It is a great achievement to open a new world, but all the more mysterious that after 50 years so much of its landscape remains unknown.

Today we ask what might determine the unseen topography and how much some recent large-data discoveries may help to map it.

The idea for this post arose from a possibly phantom memory that Juris (co-)wrote a short draft survey on “Shapes of Computations” sometime in 1986–1989 when I was at Cornell. I recall the specific phrase “long and skinny” to describe space-bounded computations. An ${\ell = O(\log n)}$ space-bounded computation can explore all of a polynomial ${p(n)}$-sized undirected graph by saving just the current node and some ${O(\log n)}$ auxiliary information by which to choose a neighbor for the next step. The trace of this computation becomes an ${\exp(\ell)}$-length sequence of ${\ell}$-bit strings. A polynomial-space computation doing an exponential search of a game tree has the same long-and-skinny “shape” even though the scale is greater with regard to the input length ${n}$. Polynomial time-bounded computations, however, can use polynomial space, whereupon they become “short and fat.” Breadth-first search of a graph is a canonical algorithm that hogs space for its relatively short duration.

Which computations fall between these “Laurel and Hardy” extremes? For SAT and the other ${\mathsf{NP}}$-complete problems, this is the great question. The surest way to separate ${\mathsf{NP}}$ from ${\mathsf{P}}$ and from ${\mathsf{PSPACE}}$ would be to characterize these problems by a third distinct shape of computation. But we have not even separated ${\mathsf{P}}$ from ${\mathsf{PSPACE}}$, nor logspace from ${\mathsf{NP}}$, so what can we pretend to know?

## Shapes of Proofs

My memory has probably been transferred from a column Juris wrote with his students Richard Chang, Desh Ranjan, and Pankaj Rohatgi for the May 1990 issue of the Bulletin of the EATCS. It has this nice diagram:

The analogy between computations and proofs has been instrumental since the early days of Kurt Gödel and Alonzo Church and Alan Turing. Proofs do, however, give nondeterminism “for free”; ${\mathsf{NP}}$ is treated the same as ${\mathsf{P}}$ in their diagram, ${\mathsf{NEXP}}$ same as ${\mathsf{EXP}}$, while “nondeterministic polynomial space” equals ${\mathsf{PSPACE}}$. Hence I’ve regarded “proofs” as secondary to “computations” as objects for complexity. However:

• We have a lot more examples of proofs, going back before Euclid;

• The search for a proof is computational;

• Interactive proofs reflect a two-player game setting where the emphasis is more on solving; and

• We now have examples of large computer-generated proofs to analyze.

The EATCS column defines a notion of width of a proof, characterizes ${\mathsf{PSPACE}}$ via polynomial width proofs, and marvels at how the classic interactive protocol for ${\mathsf{IP = PSPACE}}$ retains the “skinny” shape with less length. Indeed, in cases where the verifier is able directly to check evaluations of the unique multilinear extension ${\hat{f}}$ of the arithmetization ${f}$ of a quantified Boolean formula, every proof step involves just two terms ${a_i x_i + b_i}$ for the rounds ${i = 1}$ to ${n}$. A related form of skinniness had been brought out by Jin-Yi Cai and Merrick Furst in their 1987 paper, “${\mathsf{PSPACE}}$ Survives Three-Bit Bottlenecks.” The column goes on to emphasize that the ${\mathsf{IP}}$ form of the proof lends itself to quicker probabilistic verification. This aspect was shortly codified in the definition of probabilistically checkable proof, which lends itself most readily to characterize ${\mathsf{NP}}$ and ${\mathsf{NEXP}}$.

## Three Kinds of Proofs

Amid all this development on “long and skinny” proofs, what can we say about “short and fat” ones? Intuitively, such proofs have lots of cases, but that is not their full story. The network of logical dependence matters too. Hence we think there are most helpfully three kinds of proofs in regard to shape:

1. Proofs with close to one single path to the conclusion;

2. Proofs with many cases, but the cases have a simple hierarchical tree pattern and can mostly be checked sequentially;

3. Proofs with a more web-like pattern of logical dependence between parts.

Direct evaluations of quantified Boolean formulas in ${\mathsf{PSPACE}}$ have type 2, while the interactive proof with polynomials gives the “feel” of type 1 to both the prover and the recipient of the proof.

Chess problems prefer types 1 or a limited form of 2 for esthetics. The website ChessBase.com recently republished the longest branching-free “Mate-In-N” problem ever created, by Walther Jörgenson in 1976. It is mate-in-203 with no alternative move allowed to the winner, and virtually no case analysis of alternate defensive tries by the loser either.

However, a chess search often has type 3. Often there will be different starting sequences of moves that come to the same position ${P}$. The value of ${P}$ that was computed the first time is stored in a hash table so that the later sequences are immediately shown that value, cutting off their need for any further work. This resembles breadth-first search insofar as marked nodes may be touched later along other paths. The dependencies of values become web-like. Injured values from hash collisions can cause huge ripple effects, as I covered in a post three years ago.

The stored-hash tactic is much the same as a lemma in a proof that is used multiple times. We suspect that last year’s 6-gigabyte computer-generated proof of discrepancy > 2 in the Erdős Discrepancy Conjecture has many such lemmas, and hence is more type 3 than 2. The provers Boris Konev and Alexei Lisitsa have an updated page with further computations. They do not link the whole impossibility proof of a length 1,161-or-more sequence of discrepancy 2, but do give some of the “DRUP” certificates of unsatisfiability. DRUP stands for reverse-unit propagation with clause deletion, and that propagation strikes us as largely composed of cases and lemmas. The subcases might be codable at high level via predicates ${P(m,\cdots)}$ for m < 1,161 expressing the unavailability of length-${m}$ subsequences fulfilling some extra conditions ${(\cdots)}$, with such predicates being copiously re-used.

One finds an explosion of stored sub-cases in chess endgame tables. However, in many positions a vast majority of them are to prove wins that a chess master would see as “trivially true” in a few seconds. In other cases an alternative by the loser might simply jump to a later stage of the main trunk line, thus merely accelerating the same loss rather than posing a new case. (Similarly, an alternative by the winner might merely allow the defender to wind back to an initial stage, without much need for separate verification.) We wonder how far this pertains to the longest endgame win discovered in the new 7-piece endgame tables–mate in 545 moves. That is, how large is the part of the proof tree that needs to be specified, so that a chess program given values for positions in the tree could verify the rest via its local search?

## Computations Short-Cutting Proofs?

This last eventuality prompts our new speculation: Perhaps we can rigorously develop a science of when large sections of proofs can be effectively handwaved. This would appeal to the computational style most often postulated for our human brains: shallow but broad and with great power of heuristic association, analogous to the complexity class ${\mathsf{TC^0}}$ of poly-size constant-depth threshold circuits. Even when yoked with computers the added-value of our grey matter is not to be discounted, as attested in my joint paper last summer—in section 6 showing how human-computer tandems performed better at chess in 2005–08 than computers alone.

We have recently twice covered a conjecture by Freeman Dyson that one feels should lend itself to this kind of treatment. Many other open conjectures in number theory are felt to be “probably” true, where “probably” has a technical sense that might be developed further into some kind of dependent structure: if that handwave is valid then all-the-more certainly so is this one. The idea could be helped by enumeration of exceptions that, once handled, enable the rest of the proof to be executed with a broad brush. As linked from an essay post by Scott Aaronson, Tim Gowers relates a relevant opinion by Don Zagier in a MathOverflow comment. We morph this opinion to say that mathematicians may need a “handwave heuristic” simply because many “obviously true” statements don’t connect to anything substantial that would give a reason for their truth.

This could push many proofs of type 3 toward types 2 or 1. Note that in the way interaction and randomness combine to move type 2 toward type 1, we are already agreeing to tolerate a chance of error. It is the nature of the kind of error involved in converting instances of type 3 into type 2 that needs further consideration. We wonder whether current developments such as homotopy type theory are embracing not just exact patterns but also heuristics for when a search is overwhelmingly likely to succeed—or to fail.

This still leaves our original question of shapes of computations. In the past both Dick and I have undertaken some exploration of conditions under which computations might be similarly “self-improved.” That idea will have to be for another time.

## Open Problems

Can we assign particular “shapes” of computations canonically to specific computational problems? Can this help guide concrete attacks, or is it no more than tantamount to solving the big open questions to begin with?

Again we congratulate Juris and Richard on this 50th anniversary of their achievement. We also tip our hat to a comment by John Sidles in our “Case Against Cases” post which partly prompted this further essay.

[word changes to second paragraph of “short-cutting” section, ending, and caption]

The polynomial hierarchy is infinite for a random oracle

Benjamin Rossman, Rocco Servedio, and Li-Yang Tan have made a breakthrough in proving lower bounds on constant-depth circuits. It came from a bi-coastal collaboration of Rossman visiting the Berkeley Simons Institute from Japan and Tan visiting from Berkeley to Servedio at Columbia University in New York. Their new paper solves several 20- and 30-year old open problems.

Today we congratulate them on their achievement and describe part of how their new result works.

Problems beyond brute force search

 Cropped from Wikipedia source

Hans-Joachim Bremermann was a mathematician and biophysicist. He is famous for a limit on computation, Bremermann’s limit, which is the maximum computational speed of a self-contained system in the material universe.

Today Ken and I wish to talk about the limit and why it is not a limit.

 “Dr. Kibzwang” source

Colin Potts is Vice Provost for Undergraduate Education at Georgia Tech. His job includes being a member of the President’s Cabinet—our president, not the real one—and he is charged with academic policies and changes to such policies. He is also a College of Computing colleague and fellow chess fan.

Today I want to state a conjecture about the behavior of faculty that arose when Tech tried to change a policy.

How to avoid too many cases at least some of the time

Theon of Alexandria was history’s main editor of Euclid’s Elements.

Today I want to talk about case analysis proofs.

Congratulations to John Nash and Louis Nirenberg on the 2015 Abel Prize

 Combined from src1, src2.

John Nash and Louis Nirenberg have jointly won the 2015 Abel Prize for their work on partial differential equations (PDEs). They did not write any joint papers, but Nirenberg evidently got Nash excited about David Hilbert’s 19th problem during Nash’s frequent visits to New York University’s Courant Institute in the mid-1950s. Nash in return stimulated Nirenberg by his verbal approach of barraging a problem with off-kilter ideas. The Norwegian Academy of Sciences and Letters recognized their ‘great influence on each other’ in its prize announcement.

Today we congratulate both men on their joint achievement.

More mileage than expected from a little example

 Cropped from World Science Festival source

Sean Carroll is a cosmologist in the Department of Physics at Caltech. He also maintains a blog, “Preposterous Universe,” and writes books promoting the public understanding of science. I have recently been enjoying his 2010 book From Eternity to Here: The Quest for the Ultimate Theory of Time.

Today—yes, Carroll would agree that there is a today—I would like to share an interpretation of a little quantum computing example that occurred to me while reading his book.


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Problem 84

When ethane $\left(\mathrm{C}_{2} \mathrm{H}_{6}\right)$ reacts with chlorine $\left(\mathrm{Cl}_{2}\right),$ the main product is $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}$ , but other products containing $\mathrm{Cl},$ such as $\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{Cl}_{2},$ are also obtained in small quantities. The formation of these other products reduces the yield of $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}$ (a) Calculate the theoretical yield of $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}$ when 125 $\mathrm{g}$ of $\mathrm{C}_{2} \mathrm{H}_{6}$ reacts with 255 $\mathrm{g}$ of $\mathrm{Cl}_{2},$ assuming that $\mathrm{C}_{2} \mathrm{H}_{6}$ and $\mathrm{Cl}_{2}$ react only to form $\mathrm{C}_{2} \mathrm{H}_{2} \mathrm{Cl}$ and $\mathrm{HCl} .(\mathbf{b})$ Calculate the percent yield of $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}$ if the reaction produces 206 $\mathrm{g}$ of $\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{Cl}$

(a) SEE SOLUTION
(b) SEE SOLUTION

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Video Transcript

Okay, So in this problem, we have to find out two things. First, we have to calculate the theoretical yield auf sie to H five seal that was produced in the creation ofthe See to it six and sealed your glowing Yes. And then we have to find out the person yield off secret trip seal if the reaction produces two or six gram ofthe situation. So our first it will be we'll have to write down the reaction equation and from there we will have to go find out how many most off to see to it. Six is reacting with how many months off chlorine to propose How many? Most ofthe situation ripe seal and from there since they were given the mass ofthe situate, seas and seal to reacting from these two men as we can find out the number of malls of situate six and sealed reacting And from that we can find out the number of most of city which five seal being produced were from where we can find out, the mass of civilians were still produced. So it's started writing down the reaction equation here. So the reaction is between see to it seeks and seal two, which will give us see to H five seal plus eight seal now. So first we have to make sure that if the reaction is balanced because if variation is not balanced, will not be able to know how many moles off which react and Israel is reacting with how many, most of the other and how many moles of the product is being produced here. Okay, so from here we can just see that the reaction is balanced with our bare eyes because we have two carbon here to carbon. Here we have six had risen in the reactor side. We have five plus one. Six had risen and we have two crew in the reactor inside and one want to clone in the products. So the reaction equation is balanced. That means one more off secret six will recreate one mole off chlorine gas two produced one more lof situation seal and one more off. It's still now, since we have to find out on the mass off this product here first you have to know how many. Most ofthe situate six is reacting with. How many most ofthe sale, though, and we can find out that from the mass off these two given in our, um in the question. So in the question, we're given that the mass ofthe C two eight six reacting is one twenty five gram that miss you can find out the number ofthe malls off secret six, which is it's mass over the molar mass of the massive city it seeks. A reacting here is one twenty five gram and the monuments of secret secret is thirty point or seven gram, which is simply the momus are some of the more amounts of carbon and had risen, so seizing up to carbon, it will be so Carmen smaller manages to observe two times to exist twenty four and federal, a smaller masses. One point always seeks hydrogen will give us six months something. And the sun when he was thirty one door sonogram. Now, this will give us the value ofthe four point one six small. So this is the number of moral situate six reacting here. And then you have to find out the number of matloff chlorine gas, which is reacting. So the massive growing is reacting. Here is to fifty five gram and the molar Mass of clone is seventy point nine o six gram. This will give us a bell off three point six zero mall. So this is the number of matloff chlorine reacting. So if we just put this values ear, that man's for Plant one six, small six six six, reacting with three point six zero mall chlorine. And since for American see that one more off each reactant will react to produce one more of the product that men's from here. We have to find out the limiting re attend, and the and the number of Mall of There of the limiting Regent will be the number of mall off citizens for still being produced here. So for American easily see that since one one of each reacting will react. And so in this case, the limiting regent is It's chlorine because three point six zero mole off chlorine will react with Trooper six, Animal ofthe city six and then some excess off this situation, six will be remaining in the solution. And when all the chlorine is consumed, the relational stop And as you know, that the reactant which is I'm completely consumed first is the limiting region that Mr Limiting Reason here is chlorine, and the product which will be produced is equal to the number ofthe mall of the limiting region here because one more of the limiting Regent will produce formal of the product that mystery parent six. Small off going to reproduce three point six more. Obviously. So now we know the number of Mall of situations were still being produced. And from here we can find out the massive cities for seal produced. What is very busy. As you know, that mass is equal to number off malls, times molar mass and the number of mostly have found here is three point six zero. Lippman's the mass ofthe situation tribe seal produced will be three point six serum all times the molar mass ofthe situation absolute, which is sixty four point five one cramp our mall. And from here, this will give us two. Thirty two crumb. So this's the so this is the mass of the situation five seal, which will be which would be phones are in this reaction by and, um, my reacting one twenty five gram ofthe situation situate sees with him. If I graham off, chlorine will give us two thirty Diagram ofthe situation. Absolutely now, in our question, be we have to find out the person field of sandwich five seal produced. And that is simply ah on the actual killed, which is thiss over. Oh, no. So this is our theoretical ill. So our actually is given, as do a six gram those things crime is being produced in this actual relation. That means how are you? Percent l will be the actual field, which is two or six crumb over our theoretical ill, which we have found from the calculations, which is to thirty two gram times one hundred percent. And it will give us a fellow off eighty eight point seven nine percent. So this is the this is the person they love. Cindrich five seal per used. If two six gram off the actual ill is founded, The reaction


text:
ListDirectory - Maple Help

FileTools

 ListDirectory
 display the contents of a directory

 Calling Sequence ListDirectory(dir, opts )

Parameters

 dir - path to the directory to be displayed opts - (optional) argument of the form option or option=value where option is one of absolute, all, base, long, recurse, or returnonly

Options

 • ListDirectory accepts these optional arguments: absolute, all, base, depth, long, recurse, and returnonly.
 • absolute=truefalse
 Specifying absolute=true causes ListDirectory to return absolute paths for all listed files (see AbsolutePath). With the default value absolute=false, only the filenames themselves are returned.
 • all=truefalse
 By default, ListDirectory does not include filenames that begin with "." in its list of results. Specifying the all option causes ListDirectory to include these files.
 • base=one of bindir, datadir, homedir, mapledir, tempdir, toolboxdir, worksheetdir.
 The base option specifies a base directory. If specified, the parameter dir is interpreted as a relative path under this directory. For the meaning of worksheetdir see interface; for any other of these symbols, see kernelopts.
 • depth=posint or infinity
 Specifying depth with a value other than 1 causes ListDirectory to recursively traverse the complete directory tree starting with dir to the depth specified by depth.
 Note that if depth is set to a value other than 1, the value of option absolute is implicitly set to true, and the special directories "." and ".." will not appear in the list of results even if all=true was specified.
 Specifying depth=infinity is equivalent to specifying recurse=true.
 • long=truefalse
 Specifying long causes ListDirectory to returns a DataFrame structure which contains the name together with additional properties about each file, including size, whether or not the file is a directory, access permissions, and last modification date. The default is false.
 • recurse=truefalse
 Specifying recurse=true causes ListDirectory to recursively traverse the complete directory tree starting with dir. The default is false.
 Note that if recurse=true, the value of option absolute is set to true, and the special directories "." and ".." will not appear in the list of results even if all=true was specified.
 Specifying recurse=true is equivalent to specifying depth=infinity.
 • returnonly=string
 The returnonly option is used to select elements from the list of filenames. Only the filenames that match the pattern specified in the string are returned. The pattern must conform to the form defined by StringTools[WildcardMatch].

Description

 • The ListDirectory(dir) calling sequence returns the contents of the directory dir as a list.
 • If dir does not exist, ListDirectory generates an error.
 • If dir is not a directory, ListDirectory generates an error.
 • For information on filenames, see file.

Examples

 > $\mathrm{DataDirectory}≔\mathrm{FileTools}\left[\mathrm{JoinPath}\right]\left(\left["data"\right],\mathrm{base}=\mathrm{mapledir}\right)$
 ${\mathrm{DataDirectory}}{≔}{"/maple/cbat/active/169462/data"}$ (1)
 > $\mathrm{FileTools}\left[\mathrm{ListDirectory}\right]\left(\mathrm{DataDirectory}\right)$
 $\left[{"CAD"}{,}{"Excel"}{,}{"FMUs"}{,}{"GroupData"}{,}{"SQLite"}{,}{"Start_el.mw"}{,}{"Start_es.mw"}{,}{"Start_fr.mw"}{,}{"Start_hu.mw"}{,}{"Start_ko.mw"}{,}{"Start_pt_BR.mw"}{,}{"Start_zh_CN.mw"}{,}{"Start_zh_TW.mw"}{,}{"ThermophysicalData"}{,}{"Workbook"}{,}{"assistants"}{,}{"audio"}{,}{"datasets"}{,}{"dsolve"}{,}{"eBookTools"}{,}{"example"}{,}{"finance"}{,}{"help"}{,}{"images"}{,}{"ks.txt"}{,}{"plotting"}{,}{"portal"}{,}{"q"}{,}{"sigpro"}{,}{"stylesets"}{,}{"tutors"}{,}{"videos"}{,}{"xls"}{,}{"xml"}{,}{"xsl"}{,}{"zoneinfo"}{,}{"StudyGuides"}{,}{"Start.mw"}\right]$ (2)
 > $\mathrm{FileTools}\left[\mathrm{ListDirectory}\right]\left(\mathrm{DataDirectory},'\mathrm{all}'\right)$
 $\left[{"."}{,}{".."}{,}{"CAD"}{,}{"Excel"}{,}{"FMUs"}{,}{"GroupData"}{,}{"SQLite"}{,}{"Start_el.mw"}{,}{"Start_es.mw"}{,}{"Start_fr.mw"}{,}{"Start_hu.mw"}{,}{"Start_ko.mw"}{,}{"Start_pt_BR.mw"}{,}{"Start_zh_CN.mw"}{,}{"Start_zh_TW.mw"}{,}{"ThermophysicalData"}{,}{"Workbook"}{,}{"assistants"}{,}{"audio"}{,}{"datasets"}{,}{"dsolve"}{,}{"eBookTools"}{,}{"example"}{,}{"finance"}{,}{"help"}{,}{"images"}{,}{"ks.txt"}{,}{"plotting"}{,}{"portal"}{,}{"q"}{,}{"sigpro"}{,}{"stylesets"}{,}{"tutors"}{,}{"videos"}{,}{"xls"}{,}{"xml"}{,}{"xsl"}{,}{"zoneinfo"}{,}{"StudyGuides"}{,}{"Start.mw"}\right]$ (3)
 > $\mathrm{FileTools}\left[\mathrm{ListDirectory}\right]\left(\mathrm{DataDirectory},'\mathrm{all}','\mathrm{returnonly}'="f*"\right)$
 $\left[{"finance"}\right]$ (4)
 > $\mathrm{FileTools}\left[\mathrm{ListDirectory}\right]\left(\mathrm{DataDirectory},'\mathrm{returnonly}'="*\left[ae\right]*"\right)$
 $\left[{"Excel"}{,}{"GroupData"}{,}{"SQLite"}{,}{"Start_el.mw"}{,}{"Start_es.mw"}{,}{"Start_fr.mw"}{,}{"Start_hu.mw"}{,}{"Start_ko.mw"}{,}{"Start_pt_BR.mw"}{,}{"Start_zh_CN.mw"}{,}{"Start_zh_TW.mw"}{,}{"ThermophysicalData"}{,}{"assistants"}{,}{"audio"}{,}{"datasets"}{,}{"dsolve"}{,}{"eBookTools"}{,}{"example"}{,}{"finance"}{,}{"help"}{,}{"images"}{,}{"portal"}{,}{"stylesets"}{,}{"videos"}{,}{"zoneinfo"}{,}{"StudyGuides"}{,}{"Start.mw"}\right]$ (5)
 > $\mathrm{FileTools}\left[\mathrm{ListDirectory}\right]\left(\mathrm{DataDirectory},'\mathrm{all}','\mathrm{returnonly}'="*\left[ae\right]*"\right)$
 $\left[{"Excel"}{,}{"GroupData"}{,}{"SQLite"}{,}{"Start_el.mw"}{,}{"Start_es.mw"}{,}{"Start_fr.mw"}{,}{"Start_hu.mw"}{,}{"Start_ko.mw"}{,}{"Start_pt_BR.mw"}{,}{"Start_zh_CN.mw"}{,}{"Start_zh_TW.mw"}{,}{"ThermophysicalData"}{,}{"assistants"}{,}{"audio"}{,}{"datasets"}{,}{"dsolve"}{,}{"eBookTools"}{,}{"example"}{,}{"finance"}{,}{"help"}{,}{"images"}{,}{"portal"}{,}{"stylesets"}{,}{"videos"}{,}{"zoneinfo"}{,}{"StudyGuides"}{,}{"Start.mw"}\right]$ (6)
 > $\mathrm{FileTools}\left[\mathrm{ListDirectory}\right]\left("/home/maple/m/foo1.m"\right)$
 > $\mathrm{FileTools}\left[\mathrm{ListDirectory}\right]\left("/home/maple/noexist"\right)$

Compatibility

 • The depth, long and recurse options were introduced in Maple 2018.


text:
1. ## Partial sum convergence

Hey all, I'm having a bit of trouble on this one:

Suppose $\sum^{\infty}_{1} a_{k}$ is a series with partial sums $S_{n} = \frac{1}{2n +1}, n \in N$

a) Explain why $\sum^{\infty}_{1} a_{k}$ is convergent and find the sum of the series.

b) Find $a_{k}$, k = 1, 2, ...

Some of the scratch work I've been doing:
$S_{1} = \frac{1}{3} = a_{1}$
$S_{2} = \frac{1}{5} = a_{1} + a_{2}$
$S_{3} = \frac{1}{7} = a_{1} + a_{2} + a_{3}$ ...

$a_{1} = \frac{1}{3}$
$a_{2} = -\frac{2}{15}$
$a_{3} = -\frac{2}{35}$

$\sum^{\infty}_{1} a_{k} = \frac{1}{3} - 2(\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + ... + \frac{1}{(2n+1)(2n+3)} + ... )$

Something is obviously not jumping out at me. Any help would be appreciated.

2. The limit of the partial sums of a convergent series is the sum of the series.

3. Yeah I knew that much, but something in the back of my head was trying to tell me that that was way to easy.

Sn -> 0 right? Please tell me I'm wrong and there is actually some work to do here.

4. Unless it's a trick question, there doesn't seem to be much to say. I guess you could try and explain why a series converges iff its sequence of partial sums converges.

5. This is aggravatingly dumb. I'll ask the professor tomorrow I guess.

Alright then, so what is $a_{k}$, k = 1,2... ?

6. The solution in implicit in definition of 'partial sum' ...

$S_{n} = \sum_{k=1}^{n} a_{k}$ (1)

From (1) we derive immediately...

$a_{n} = S_{n} - S_{n-1} = \frac{1}{2n+1} - \frac{1}{2n-1} = \frac{2}{1-4n^{2}}$ (2)

The series of the $a_{n}$ is convergent and its sum is...

$S=\lim_{n \rightarrow \infty} S_{n} = 0$ (3)

Kind regards

$\chi$ $\sigma$