text: Batch size N 1 2 3 4 5 6 batch N = 1 2.264 1.224 0.8927 0.7514 0.6976 0.6571 N = 2 3.185 2.83 2.791 2.823 2.837 2.86 N = 3 9.907 10.14 10.25 10.16 10.19 10.21 N = 4 29.41 29.74 30.01 30.03 30.23 30.47 N = 5 74.48 74.39 74.16 74.01 73.88 74.32 N = 6 173.4 173.9 175.2 171 170.5 172.4 N = 7 329.4 330.4 329.6 331 (a) Weighted projection matrix Pw Batch size N 1 2 3 4 5 6 batch N = 1 2.382 1.26 0.8833 0.6986 0.5832 0.6534 N = 2 3.864 2.101 2.495 1.968 2.28 1.964 N = 3 7.092 6.788 6.9 6.89 6.888 6.836 N = 4 24.79 22.15 26.63 24.02 24.16 24 N = 5 70.62 76.6 61.55 56.35 56.38 58.98 N = 6 179.8 145.7 129.9 145.7 131.9 144.1 N = 7 411.6 412.7 393 510.3 − (b) Weight-adjusted projection M 1M 1/w Table 2: Runtimes (nanoseconds) per element for weighted and weight-adjusted projections as a function of batch size N . The lowest runtimes are highlighted in bold. batch to the right hand side. Since the weight w is spatially varying and distinct from element-to-element, we pre- compute and store P explicitly over each element prior to time-stepping. The weight-adjusted DG method w − − is equivalent to replacing the matrices M 1M with the weight-adjusted projection matrix M 1M , w 1/w which can be applied in a matrix-free fashion as described in Section . We now examine computational costs associated with the use of the weight-adjusted DG method. Com- putational statistics are computed using the Nvidia proﬁler nvprof. We consider ﬁrst the costs associated with the use of weight-adjusted approximations to scalar weighted mass matrices. While WADG clearly re- duces storage costs associated with high order DG methods, it is less clear how WADG aﬀects computational runtime on accelerator and many-core architectures. We compare the application of pre-computed and stored weighted projection matrices P with a matrix- − w free application of the weight-adjusted matrix M 1M using a GPU-accelerated implementation. As ≥ 1/w described in [11, 2], we batch process N 1 elements within a single kernel workgroup. Table 2 displays batch the average runtime per element for the weighted and weight-adjusted projection kernels as a function of batch size N when using a mesh of 50, 000 elements and a quadrature which is exact for polynomials of batch degree 2N + 1. When processing only a single element per batch, WADG is less eﬃcient than weighted projection at all orders. However, the cost of WADG goes down rapidly with the number of elements per batch. After optimizing over the batch size, weight-adjusted projection is faster than weighted projection up to N = 6. At N = 7, WADG is slower than weighted projection, as the batch size is limited by the maximum number of active threads.4 We note that, if the strength of quadrature is reduced from 2N + 1 to 2N, the resulting − × WADG runtimes are faster than weighted projection at all tested orders, achieving between a 1.5 2.3 speedup for N = 1, . . . , 7 while maintaining virtually identical numerical results [1]. We can take a closer look at these results using the Nvidia proﬁler nvprof, looking in particular at the 4In our implementation, the number of active threads per workgroup is the number of quadrature points multiplied by the number of elements per batch. The batch size is limited by the maximum number of threads in a workgroup (1024 for Nvidia GPUs). For N = 7, the quadrature rule of degree 2N + 1 contains 214 points; the largest batch size we can run is then 4 elements, as processing 5 elements per batch requires 1070 threads. 32 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: (cid:74) (cid:75) (cid:74) (cid:75) → ∞ non-zero jumps AT v and A σ . In fact, it was shown in [21] that, in the limit as τ , τ , the n n v σ eigenspaces of DG discretizations split into a conforming part consisting of u, σ which satisfy (cid:74) (cid:75) (cid:74) (cid:75) A u = 0, AT σ = 0 n n and an non-conforming part (deﬁned through the L2 orthogonal complement) corresponding to eigenvalues −∞ (cid:74) (cid:75) contain real parts which approach . For the linear elastic wave equations, these conditions are equivalent to requirements of C0 continuity for u and normal continuity of the stress tensor S˜ n = 0. 3.3 The semi-discrete matrix system for DG The solution to (2) can be approximated by discretizing in space and using an explicit time integrator, which require(cid:18)s only ev(cid:19)aluations o(cid:32)f local contribut(cid:33)ions over D(cid:28)k to the DG formulation (cid:29) (cid:88) (cid:74) (cid:75) (cid:74) (cid:75) d ∂v ∂σ 1 τ (cid:18) ρ , w(cid:19) = (cid:32) AT , w(cid:33) +(cid:28) AT σ + v AT A v ,(cid:29)w ∂t (cid:88) i ∂x 2 n 2 n n L2(Dk) i=1 i L2(Dk) L2(∂Dk) (cid:74) (cid:75) (cid:74) (cid:75) d − ∂σ ∂v 1 τ C 1 , q = A , q + A v + σ A AT σ , q . (4) ∂t (cid:16) (cid:17) i ∂x 2 n 2 n n L2(Dk) i=(cid:98)1 i L2(Dk) (cid:99) L2(∂Dk) { } Let φ Np be a basis for PN D .1 We deﬁne the reference mass matrix M and the physical mass i i=1 (cid:16) (cid:17) (cid:90) (cid:90) (cid:90) matrix M for an elemen(cid:99)t Dk as (cid:98) (cid:98) (cid:98) (cid:98) M = φ φ dx, (M) = φ φ dx = φ φ J dx. j i (cid:99) ij j i j i ij D Dk D For aﬃne mappings, J is cons(cid:90)tant and M = JM. We also d(cid:90)eﬁne weak di(cid:90)ﬀerentiation matrices Sk and face mass matrix M such that f (cid:98) ∂φ (cid:98) (S ) = j φ dx, (M ) = φ φ dx = φ φ Jf dx, k ij ∂x i f ij (cid:98)j i j i Dk k f f (cid:99) (cid:99) where Jf is the Jacobian of the mapping fro(cid:99)m a reference face f to f. For aﬃnely mapped simplices, Jf is also constant and M = JfM , where the deﬁnition of the reference face mass matrix M is analogous to f f f the deﬁnition of the reference mass matrix M. ∈ R ∈ R × Finally, we deﬁne weighted mass matrices. Let w(x) and W (x) m n. Then, scalar and matrix- weighted mass matrices M(cid:90) and M are deﬁned through w W     M . . . M W1,1 W1,n (Mw)ij = w(x)φj(x)φi(x) dx, MW = ... ... ... , Dk M . . . M Wm,1 Wm,n where M is the scalar weighted mass matrix weighted by the (i, j) entry of W . Wi,j Local contributions to the DG variational form may then be evaluated in a quadrature-free manner using these matrices. Let Σ , V denote vectors containing degrees of freedom for solution components σ , v such i i i i 1In our implementations, we use nodal basis functions at Warp and Blend interpolation points [29]. These basis functions are deﬁned implicitly using an orthogonal polynomial basis on the reference simplex [23]. However, we note that the implementation and formulation are independent of the speciﬁc choice of polynomial basis. 8 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: non-zero

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text: elastic

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text: reference

text: face

text: f

text: to

text: f.

text: For

text: aﬃnely

text: mapped

text: simplices,

text: Jf

text: is

text: also

text: constant

text: and

text: M

text: JfM

text: where

text: the

text: deﬁnition

text: of

text: the

text: reference

text: face

text: mass

text: matrix

text: M

text: is

text: analogous

text: to

text: f

text: f

text: f

text: the

text: deﬁnition

text: of

text: the

text: reference

text: mass

text: matrix

text: M.

text: R

text: R

text: Finally,

text: we

text: deﬁne

text: weighted

text: mass

text: matrices.

text: Let

text: w(x)

text: and

text: W

text: (x)

text: m

text: n.

text: Then,

text: scalar

text: and

text: matrix-

text: weighted

text: mass

text: matrices

text: and

text: M

text: are

text: deﬁned

text: through

text: w

text: W

text: M

text: M

text: (Mw)ij

text: w(x)φj(x)φi(x)

text: dx,

text: MW

text: Dk

text: M

text: M

text: Wm,n

text: where

text: M

text: is

text: the

text: scalar

text: weighted

text: mass

text: matrix

text: weighted

text: by

text: the

text: (i,

text: j)

text: entry

text: of

text: W

text: Wi,j

text: Local

text: contributions

text: to

text: the

text: DG

text: variational

text: form

text: may

text: then

text: be

text: evaluated

text: in

text: a

text: quadrature-free

text: manner

text: using

text: these

text: matrices.

text: Let

text: Σ

text: V

text: denote

text: vectors

text: containing

text: degrees

text: of

text: freedom

text: for

text: solution

text: components

text: σ

text: v

text: such

text: i

text: i

text: i

text: i

text: our

text: implementations,

text: we

text: use

text: nodal

text: basis

text: functions

text: at

text: Warp

text: and

text: Blend

text: interpolation

text: points

text: These

text: basis

text: functions

text: are

text: deﬁned

text: implicitly

text: using

text: an

text: orthogonal

text: polynomial

text: basis

text: on

text: the

text: reference

text: simplex

text: However,

text: we

text: note

text: that

text: the

text: implementation

text: and

text: formulation

text: are

text: independent

text: of

text: the

text: speciﬁc

text: choice

text: of

text: polynomial

text: basis.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 4 We have simulated a number of observations for a∗ = ACKNOWLEDGMENTS 0.95 and diﬀerent values of the parameters L, of the view- ing angle i, and with diﬀerent emissivity proﬁles. The simulations have been treated as real data and ﬁtted with XSPEC2, employing the model RELLINE for the iron line [41]. If L is not very large, the ﬁts are usually acceptable, in the sense that there are not unresolved features and the minimum of the reduced χ2 is not much higher than 1. If we require that the analysis of our sim- ulations should provide spin measurements a∗ > 0.9, we ﬁnd that a conservative constraint for L is L/M < 1.2. More stringent constraints can be deﬁnitively obtained with a more detailed analysis, which is beyond the ex- plorative work presented in this work. V. CONCLUDING REMARKS In this paper, we have discussed how to test conformal gravity with observational data of astrophysical black holes. In particular, we have considered the singularity- free rotating black hole solution in Eq. (2) and obtained a rough constraint on the parameter L. The latter can change the structure of the accretion disk around black holes and, in particular, can alter the ISCO radius, a quantity that can have a strong impact on the X-ray spectrum of black holes. Within a very simple analysis, we have discussed the expected iron line in the reﬂection spectrum of these spacetimes and shown that current ob- servations require L/M < 1.2. For higher values of L, it is not possible to reproduce the extended low energy tail in the iron line as observed in the spectra of several sources. We would like to remark that our analysis is quite sim- ple, but the constraint L/M < 1.2 is robust and is not aﬀected by our assumptions on the astrophysical model. In particular, in typical spin measurements or tests of the Kerr metric, a crucial point is the emissivity proﬁle, namely the ﬂux of the reﬂection spectrum at diﬀerent radii. Changing the emissivity proﬁle, we alter the rel- ative contribution from diﬀerent parts of the accretion disk. Our result does not depend on the intensity proﬁle, because the key-point of the constraint L/M < 1.2 is that observations show broad iron lines with a low energy tail ∼ up to 1 keV, while for suﬃciently high values of L/M we cannot have very redshifted photons at any radius of the accretion disk. The photon redshift is only deter- mined by the background metric, and it is not possible to increase the redshift factor of the photons by chang- ing the astrophysical model. A more detailed analysis with the more sophisticated model presented in Ref. [42] can study real data of speciﬁc sources and get stronger constraints. We postpone such an analysis to a future work. This work was supported by the NSFC (grants U1531117 2 XSPEC is an X-ray spectral-ﬁtting software com- monly used in X-ray astronomy. See [40] and and 11305038) and the Thousand Young Talents Pro- http://heasarc.gsfc.nasa.gov/docs/xanadu/xspec/index.html gram. C.B. also acknowledges the support from the for more details. Alexander von Humboldt Foundation. ##LTLine##

text: We

text: have

text: simulated

text: a

text: number

text: of

text: observations

text: for

text: a∗

text: ACKNOWLEDGMENTS

text: and

text: diﬀerent

text: values

text: of

text: the

text: parameters

text: L,

text: of

text: the

text: view-

text: ing

text: angle

text: i,

text: and

text: with

text: diﬀerent

text: emissivity

text: proﬁles.

text: The

text: simulations

text: have

text: been

text: treated

text: as

text: real

text: data

text: and

text: ﬁtted

text: with

text: employing

text: the

text: model

text: RELLINE

text: for

text: the

text: iron

text: line

text: If

text: L

text: is

text: not

text: very

text: large,

text: the

text: ﬁts

text: are

text: usually

text: acceptable,

text: in

text: the

text: sense

text: that

text: there

text: are

text: not

text: unresolved

text: features

text: and

text: the

text: minimum

text: of

text: the

text: reduced

text: χ2

text: is

text: not

text: much

text: higher

text: than

text: If

text: we

text: require

text: that

text: the

text: analysis

text: of

text: our

text: sim-

text: ulations

text: should

text: provide

text: spin

text: measurements

text: a∗

text: we

text: ﬁnd

text: that

text: a

text: conservative

text: constraint

text: for

text: L

text: is

text: L/M

text: More

text: stringent

text: constraints

text: can

text: be

text: deﬁnitively

text: obtained

text: with

text: a

text: more

text: detailed

text: analysis,

text: which

text: is

text: beyond

text: the

text: ex-

text: plorative

text: work

text: presented

text: in

text: this

text: work.

text: V.

text: CONCLUDING

text: REMARKS

text: In

text: this

text: paper,

text: we

text: have

text: discussed

text: how

text: to

text: test

text: conformal

text: gravity

text: with

text: observational

text: data

text: of

text: astrophysical

text: black

text: holes.

text: In

text: particular,

text: we

text: have

text: considered

text: the

text: singularity-

text: free

text: rotating

text: black

text: hole

text: solution

text: in

text: Eq.

text: and

text: obtained

text: a

text: rough

text: constraint

text: on

text: the

text: parameter

text: L.

text: The

text: latter

text: can

text: change

text: the

text: structure

text: of

text: the

text: accretion

text: disk

text: around

text: black

text: holes

text: and,

text: in

text: particular,

text: can

text: alter

text: the

text: ISCO

text: radius,

text: a

text: quantity

text: that

text: can

text: have

text: a

text: strong

text: impact

text: on

text: the

text: X-ray

text: spectrum

text: of

text: black

text: holes.

text: Within

text: a

text: very

text: simple

text: analysis,

text: we

text: have

text: discussed

text: the

text: expected

text: iron

text: line

text: in

text: the

text: reﬂection

text: spectrum

text: of

text: these

text: spacetimes

text: and

text: shown

text: that

text: current

text: ob-

text: servations

text: require

text: L/M

text: For

text: higher

text: values

text: of

text: L,

text: it

text: is

text: not

text: possible

text: to

text: reproduce

text: the

text: extended

text: low

text: energy

text: tail

text: in

text: the

text: iron

text: line

text: as

text: observed

text: in

text: the

text: spectra

text: of

text: several

text: sources.

text: We

text: would

text: like

text: to

text: remark

text: that

text: our

text: analysis

text: is

text: quite

text: sim-

text: ple,

text: but

text: the

text: constraint

text: L/M

text: is

text: robust

text: and

text: is

text: not

text: aﬀected

text: by

text: our

text: assumptions

text: on

text: the

text: astrophysical

text: model.

text: In

text: particular,

text: in

text: typical

text: spin

text: measurements

text: or

text: tests

text: of

text: the

text: Kerr

text: metric,

text: a

text: crucial

text: point

text: is

text: the

text: emissivity

text: proﬁle,

text: namely

text: the

text: ﬂux

text: of

text: the

text: reﬂection

text: spectrum

text: at

text: diﬀerent

text: radii.

text: Changing

text: the

text: emissivity

text: proﬁle,

text: we

text: alter

text: the

text: rel-

text: ative

text: contribution

text: from

text: diﬀerent

text: parts

text: of

text: the

text: accretion

text: disk.

text: Our

text: result

text: does

text: not

text: depend

text: on

text: the

text: intensity

text: proﬁle,

text: because

text: the

text: key-point

text: of

text: the

text: constraint

text: L/M

text: is

text: that

text: observations

text: show

text: broad

text: iron

text: lines

text: with

text: a

text: low

text: energy

text: tail

text: up

text: to

text: keV,

text: while

text: for

text: suﬃciently

text: high

text: values

text: of

text: L/M

text: we

text: cannot

text: have

text: very

text: redshifted

text: photons

text: at

text: any

text: radius

text: of

text: the

text: accretion

text: disk.

text: The

text: photon

text: redshift

text: is

text: only

text: deter-

text: mined

text: by

text: the

text: background

text: metric,

text: and

text: it

text: is

text: not

text: possible

text: to

text: increase

text: the

text: redshift

text: factor

text: of

text: the

text: photons

text: by

text: chang-

text: ing

text: the

text: astrophysical

text: model.

text: A

text: more

text: detailed

text: analysis

text: with

text: the

text: more

text: sophisticated

text: model

text: presented

text: in

text: Ref.

text: can

text: study

text: real

text: data

text: of

text: speciﬁc

text: sources

text: and

text: get

text: stronger

text: constraints.

text: We

text: postpone

text: such

text: an

text: analysis

text: to

text: a

text: future

text: work.

text: This

text: work

text: was

text: supported

text: by

text: the

text: NSFC

text: (grants

text: XSPEC

text: is

text: an

text: X-ray

text: spectral-ﬁtting

text: software

text: com-

text: monly

text: used

text: in

text: X-ray

text: astronomy.

text: See

text: and

text: and

text: and

text: the

text: Thousand

text: Young

text: Talents

text: Pro-

text: gram.

text: C.B.

text: also

text: acknowledges

text: the

text: support

text: from

text: the

text: for

text: more

text: details.

text: Alexander

text: von

text: Humboldt

text: Foundation.

text: ##LTLine##


text: where κ = 8π and has the dimension of the inverse of a length square. sectioncolor II. BASIC EQUATIONS (cid:90) (cid:20) (cid:21) The Action for modiﬁed Gauss-Bonnet gravity reads equationcolor √ − R G S = d4x g + f( ) + S (gµν, ψ) , (5) 2κ M where S (gµν, ψ) is the matter action and ψ collectively denotes the matter ﬁelds. The Gauss-Bonnet invariant is M deﬁned as equationcolor G ≡ − R2 4R Rµν + R Rµναβ. (6) µν µναβ Varying the action with respect to the metric yields the ﬁeld equations: (cid:2) (cid:3) equationcolor − − − ∇ ∇ (cid:48) (cid:48) − G + 8 R + R g R g R g + R g + R(g g g g ) ρ σf + (f f)g = κT , (7) µν µρνσ ρν σµ ρσ νµ µν σρ µσ νρ 2 µν σρ µσ νρ µν µν − where the prime represnts the derivative with respect to and G = R (1/2)Rg is the Einstein-tensor. µν µν µν The matter energy-momentum tensor is deﬁned as usual as equationcolor √ − −√2 δ( g L ) T = − m . (8) µν g δ(gµν) (cid:20) (cid:21) Our treatment will consider Friedmann Lemaître Robertson Walker (FLRW) metric: equationcolor − dr2 ds2 = dt2 + a2(t) − + r2(dθ2 + sin2 θdφ2) , (9) 1 kr2 where a is the scale factor and≤k the≤spatial curvature. We also assume that the cosmic ﬂuid is a prefect ﬂuid with equation of state p = wµ with 0 w 1, where µ(cid:18)and p, re(cid:19)spectively are the energy density and pressure measured by the co-moving observer, and where we assume w to be constant. In this metric, the Gauss-Bonnet invariant is equationcolor k = 24 H2 + (H˙ + H2) (10) a2 where H = a˙ is the Hubble factor. This expression is particularly important because it connects the sign of to the sign of the decaeleration factor. For k ≥ 0, any cosmology which transits between acceleration and deceleration will have to change the sign of . If k < 0, this is not necessarily true. In this respect Gauss-Bonnet cosmology depends in an even more crucial way on the spatial curvature than GR. sectioncolor III. THE DYNAMICAL SYSTEMS APPROACH. G conFsitrastn,tswaepipnetraordinugceinathcoenascttainotnχm0osdu(cid:90)GchB(cid:8)athctaitonthteo(cid:0)porbotda(cid:1)uincts Rχ0 and (cid:9)χ20 are dimensionless and we redeﬁne all the equationcolor √ G − S = χ R + f χ2 + L (g , Φ) g d4x (11) 0 0 m µν where now all the constants (with the exception of χ ) in the action are assumed dimensionless. 0 Now deﬁning the parameters equationcolor ... H˙ H¨ − H˙ 2 H H˙ 3 − H˙ H¨ q = , j = , s = + 3 4 . (12) H2 H3 H4 H4 H6 H5 the cosmological equations can be written as ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: κ

text: 8π

text: and

text: has

text: the

text: dimension

text: of

text: the

text: inverse

text: of

text: a

text: length

text: square.

text: sectioncolor

text: II.

text: BASIC

text: EQUATIONS

text: The

text: Action

text: for

text: modiﬁed

text: Gauss-Bonnet

text: gravity

text: reads

text: equationcolor

text: R

text: G

text: S

text: g

text: f(

text: S

text: (gµν,

text: ψ)

text: 2κ

text: M

text: where

text: S

text: (gµν,

text: ψ)

text: is

text: the

text: matter

text: action

text: and

text: ψ

text: collectively

text: denotes

text: the

text: matter

text: ﬁelds.

text: The

text: Gauss-Bonnet

text: invariant

text: is

text: M

text: deﬁned

text: as

text: equationcolor

text: G

text: Rµν

text: R

text: Rµναβ.

text: µν

text: µναβ

text: Varying

text: the

text: action

text: with

text: respect

text: to

text: the

text: metric

text: yields

text: the

text: ﬁeld

text: equations:

text: equationcolor

text: G

text: R

text: R

text: g

text: R

text: g

text: R

text: g

text: R

text: g

text: R(g

text: g

text: g

text: g

text: ρ

text: σf

text: (f

text: f)g

text: κT

text: µν

text: µρνσ

text: ρν

text: σµ

text: ρσ

text: νµ

text: µν

text: σρ

text: µσ

text: νρ

text: µν

text: σρ

text: µσ

text: νρ

text: µν

text: µν

text: where

text: the

text: prime

text: represnts

text: the

text: derivative

text: with

text: respect

text: to

text: and

text: G

text: R

text: is

text: the

text: Einstein-tensor.

text: µν

text: µν

text: µν

text: The

text: matter

text: energy-momentum

text: tensor

text: is

text: deﬁned

text: as

text: usual

text: as

text: equationcolor

text: −√2

text: δ(

text: g

text: L

text: T

text: m

text: µν

text: g

text: δ(gµν)

text: Our

text: treatment

text: will

text: consider

text: Friedmann

text: Lemaître

text: Robertson

text: Walker

text: (FLRW)

text: metric:

text: equationcolor

text: where

text: a

text: is

text: the

text: scale

text: factor

text: and≤k

text: the≤spatial

text: curvature.

text: We

text: also

text: assume

text: that

text: the

text: cosmic

text: ﬂuid

text: is

text: a

text: prefect

text: ﬂuid

text: with

text: equation

text: of

text: state

text: p

text: wµ

text: with

text: w

text: where

text: p,

text: are

text: the

text: energy

text: density

text: and

text: pressure

text: measured

text: by

text: the

text: co-moving

text: observer,

text: and

text: where

text: we

text: assume

text: w

text: to

text: be

text: constant.

text: In

text: this

text: metric,

text: the

text: Gauss-Bonnet

text: invariant

text: is

text: equationcolor

text: k

text: (H˙

text: where

text: H

text: a˙

text: is

text: the

text: Hubble

text: factor.

text: This

text: expression

text: is

text: particularly

text: important

text: because

text: it

text: connects

text: the

text: sign

text: of

text: to

text: the

text: sign

text: of

text: the

text: decaeleration

text: factor.

text: For

text: k

text: any

text: cosmology

text: which

text: transits

text: between

text: acceleration

text: and

text: deceleration

text: will

text: have

text: to

text: change

text: the

text: sign

text: of

text: If

text: k

text: this

text: is

text: not

text: necessarily

text: true.

text: In

text: this

text: respect

text: Gauss-Bonnet

text: cosmology

text: depends

text: in

text: an

text: even

text: more

text: crucial

text: way

text: on

text: the

text: spatial

text: curvature

text: than

text: GR.

text: sectioncolor

text: III.

text: THE

text: DYNAMICAL

text: SYSTEMS

text: APPROACH.

text: G

text: and

text: are

text: dimensionless

text: and

text: we

text: redeﬁne

text: all

text: the

text: equationcolor

text: G

text: S

text: χ

text: R

text: f

text: χ2

text: L

text: (g

text: Φ)

text: g

text: m

text: µν

text: where

text: now

text: all

text: the

text: constants

text: (with

text: the

text: exception

text: of

text: χ

text: in

text: the

text: action

text: are

text: assumed

text: dimensionless.

text: Now

text: deﬁning

text: the

text: parameters

text: equationcolor

text: H˙

text: H¨

text: H˙

text: H

text: H˙

text: H˙

text: H¨

text: q

text: j

text: s

text: the

text: cosmological

text: equations

text: can

text: be

text: written

text: as

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


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text: equationcolor

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text: a 15 10 5 t 1 2 3 4 Figure 1: captioncolorNumerical solution of equation SolA. The constants H have all been chosen to be one and the initial i condition is a(0) = 0.01. a 40 30 20 10 t 5 10 15 20 Figure 2: captioncolorNumerical solution of equation SolB. The constants H have all been chosen to be one and the initial i condition is a(0) = 0.01. a 2.5 2.0 1.5 1.0 0.5 t 0.5 1.0 1.5 2.0 2.5 3.0 3.5 − Figure 3: captioncolorNumerical solution of equation SolH. The constants H have all been chosen in such a way that 4H H i 0 2 H2 > 0 and the initial condition is a(0) = 0.01. The solution presents a ﬁnite time singularity. If the above condition is violated 1 the solution approaches a static universe as in the case of SolA an SolB. ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: Comprehensive study on band-gap variations in sp3-bonded semiconductors: roles of ∗ electronic states ﬂoating in internal space Yu-ichiro Matsushita1 and Atsushi Oshiyama1 (Dated: May 29, 2020) We have performed electronic structure calculations to explore the band-gap dependence on poly- types for sp3-bonded semiconducting materials, i.e., SiC, AlN, BN, GaN, Si, and diamond. In this comprehensive study, we have found that band-gap variation depending on polytypes is common in sp3-bonded semiconductors; SiC, AlN, and BN exhibit smallest band gaps in 3C structure, whereas diamond does in 2H structure. We have also clariﬁed that the microscopic mechanism of the band- gap variations is attributed to peculiar electron states floating in internal channel space at the conduction-band minimum (CBM), and that internal channel length and the electro-static potential in channel aﬀect the energy level of CBM. INTRODUCTION The development of the modern society has been mostly attributed to semiconductor technology. In most semiconductors from elemental to compound, each atom forms sp3 bonds and take four-fold coordinated tetrahe- dral structure. It is well known that the crystal structure consisting of sp3 bonds exhibit hundreds of polytypes [1]. Their structural diﬀerence is in the stacking of the tetra- (cid:104) (cid:105) hedral units along the 111 direction in cubic structure, (cid:104) (cid:105) and 0001 direction in hexagonal structure. Zincblende and wurtzite structures are the most famous examples of them. The Zincblende structure is represented by the stacking sequence of ABC and the wurtzite by AB. FIG. 1. (Color online) Residual norms of the wavefunctions Nomenclature adopted vastly is introduced here: each of the energy bands of 3C-SiC (a) and 2H-SiC (b). The residual norms are represented by the color and the size of polytype is labeled by the periodicity of the stacking the dots. The energy of the valence-band top is set to be sequence n and the symmetry (cubic or hexagonal) such 0. The residual norm which is a measure of the ﬂoating as 2H (wurtzite), 3C (zinblende), 4H, 6H, etc. nature is calculated in the following procedure: From the { } These structural diﬀerences have been assumed to be pseudo-atomic orbitals φisolated of isolated silicon and car- i { } minor in the electronic properties. It is because there b(cid:12) on atom(cid:80)s, w(cid:12)e hav(cid:11)e (cid:10)composed o(cid:11)r(cid:12)thonormal basis set φaitom are no diﬀerences in local atomic structure up to the 2nd w(cid:12) ith the Gra(cid:12)m-Schmidt orthono(cid:12)rmalization. Then we have nearest neighbor. Valence bands consist of sp3-bonding c|alcul(cid:105)a−ted the squared res|idual norm, which is deﬁnes as φ φatom φatom φ 2 for each band n at k point. orbitals and conduction bands sp3-antibonding. How- nk i i i nk ever, it is reported that the stacking sequence aﬀects electronic properties considerably in silicon carbide (SiC) [2, 3]. SiC is indeed a manifestation of the polytypes: character. This fact is clearly shown in Fig. 1. The Dozens of polytypes of SiC are observed. Yet, surpris- ﬁgure shows the calculated residual norms of the wave- ingly, the band gaps vary by 40 %, from 2.3 eV in 3C function of 3C-SiC after projecting it to the s- and to 3.3 eV in 2H despite that the structures are locally p-atomic orbitals. While the wavefunction in valence − − identical to each other in all the polytypes [3]. This phe- bands is well described by s and p atomic orbitals, nomenon was diﬃcult to be understood in conventional conduction electrons cannot. This floating character at chemical pictures. the CBM is an important key to reveal the microscopic We have recently reported [4–6] the microscopic mech- mechanism of the band-gap variations. Channel length anism of band-gap variations in SiC polytypes based changes depending on polytypes: 3C structure has in- (cid:104) (cid:105) on the density functional theory (DFT) [7, 8]. It is ﬁnite channel length along the 110 direction, while √ found that continuum states exist in conduction bands similar channel structure is seen also in 6H structure (cid:104) (cid:105) of sp3-bonded materials, and furthermore, such peculiar with the length of about 7a /2 2 along ¯2201 which (cid:104) (cid:105) 0 electron state appears at the conduction-band minima is slanted relative to 0001 direction with a a lattice 0 (CBM) in SiC polytypes. The wavefunction at the CBM constant. Floating states at the CBM extends in the is not distributed near atomic sites, but extends (or internal space. Therefore, channel length and channel floats) in interstitial channels without atomic orbital shapes are decisive in the positions of the CBM in energy ##LTFigure## ##LTLine##

text: Comprehensive

label: title

text: study

text: on

text: band-gap

text: variations

text: in

text: semiconductors:

text: roles

text: of

text: electronic

text: states

text: ﬂoating

text: in

text: internal

text: space

text: Yu-ichiro

label: author

text: and

text: Atsushi

text: (Dated:

text: May

label: date

text: We

label: abstract

text: have

text: performed

text: electronic

text: structure

text: calculations

text: to

text: explore

text: the

text: band-gap

text: dependence

text: on

text: poly-

text: types

text: for

text: semiconducting

text: materials,

text: i.e.,

text: SiC,

text: AlN,

text: BN,

text: GaN,

text: Si,

text: and

text: diamond.

text: In

text: this

text: comprehensive

text: study,

text: we

text: have

text: found

text: that

text: band-gap

text: variation

text: depending

text: on

text: polytypes

text: is

text: common

text: in

text: semiconductors;

text: SiC,

text: AlN,

text: and

text: BN

text: exhibit

text: smallest

text: band

text: gaps

text: in

text: structure,

text: whereas

text: diamond

text: does

text: in

text: structure.

text: We

text: have

text: also

text: clariﬁed

text: that

text: the

text: microscopic

text: mechanism

text: of

text: the

text: band-

text: gap

text: variations

text: is

text: attributed

text: to

text: peculiar

text: electron

text: states

text: floating

text: in

text: internal

text: channel

text: space

text: at

text: the

text: conduction-band

text: minimum

text: (CBM),

text: and

text: that

text: internal

text: channel

text: length

text: and

text: the

text: electro-static

text: potential

text: in

text: channel

text: aﬀect

text: the

text: energy

text: level

text: of

text: CBM.

text: INTRODUCTION

text: The

text: development

text: of

text: the

text: modern

text: society

text: has

text: been

text: mostly

text: attributed

text: to

text: semiconductor

text: technology.

text: In

text: most

text: semiconductors

text: from

text: elemental

text: to

text: compound,

text: each

text: atom

text: forms

text: bonds

text: and

text: take

text: four-fold

text: coordinated

text: tetrahe-

text: dral

text: structure.

text: It

text: is

text: well

text: known

text: that

text: the

text: crystal

text: structure

text: consisting

text: of

text: bonds

text: exhibit

text: hundreds

text: of

text: polytypes

text: Their

text: structural

text: diﬀerence

text: is

text: in

text: the

text: stacking

text: of

text: the

text: tetra-

text: hedral

text: units

text: along

text: the

text: direction

text: in

text: cubic

text: structure,

text: and

text: direction

text: in

text: hexagonal

text: structure.

text: Zincblende

text: and

text: wurtzite

text: structures

text: are

text: the

text: most

text: famous

text: examples

text: of

text: them.

text: The

text: Zincblende

text: structure

text: is

text: represented

text: by

text: the

text: stacking

text: sequence

text: of

text: ABC

text: and

text: the

text: wurtzite

text: by

text: AB.

text: FIG.

text: (Color

text: online)

text: Residual

text: norms

text: of

text: the

text: wavefunctions

text: Nomenclature

text: adopted

text: vastly

text: is

text: introduced

text: here:

text: each

text: of

text: the

text: energy

text: bands

text: of

text: (a)

text: and

text: (b).

text: The

text: residual

text: norms

text: are

text: represented

text: by

text: the

text: color

text: and

text: the

text: size

text: of

text: polytype

text: is

text: labeled

text: by

text: the

text: periodicity

text: of

text: the

text: stacking

text: the

text: dots.

text: The

text: energy

text: of

text: the

text: valence-band

text: top

text: is

text: set

text: to

text: be

text: sequence

text: n

text: and

text: the

text: symmetry

text: (cubic

text: or

text: hexagonal)

text: such

text: The

text: residual

text: norm

text: which

text: is

text: a

text: measure

text: of

text: the

text: ﬂoating

text: as

text: (wurtzite),

text: (zinblende),

text: etc.

text: nature

text: is

text: calculated

text: in

text: the

text: following

text: procedure:

text: From

text: the

text: These

text: structural

text: diﬀerences

text: have

text: been

text: assumed

text: to

text: be

text: pseudo-atomic

text: orbitals

text: φisolated

text: of

text: isolated

text: silicon

text: and

text: car-

text: i

text: minor

text: in

text: the

text: electronic

text: properties.

text: It

text: is

text: because

text: there

text: on

text: basis

text: set

text: φaitom

text: are

text: no

text: diﬀerences

text: in

text: local

text: atomic

text: structure

text: up

text: to

text: the

text: ith

text: the

text: Then

text: we

text: have

text: nearest

text: neighbor.

text: Valence

text: bands

text: consist

text: of

text: the

text: squared

text: res|idual

text: norm,

text: which

text: is

text: deﬁnes

text: as

text: φ

text: φatom

text: φatom

text: φ

text: for

text: each

text: band

text: n

text: at

text: k

text: point.

text: orbitals

text: and

text: conduction

text: bands

text: How-

text: nk

text: i

text: i

text: i

text: nk

text: ever,

text: it

text: is

text: reported

text: that

text: the

text: stacking

text: sequence

text: aﬀects

text: electronic

text: properties

text: considerably

text: in

text: silicon

text: carbide

text: (SiC)

text: SiC

text: is

text: indeed

text: a

text: manifestation

text: of

text: the

text: polytypes:

text: character.

text: This

text: fact

text: is

text: clearly

text: shown

text: in

text: Fig.

text: The

text: Dozens

text: of

text: polytypes

text: of

text: SiC

text: are

text: observed.

text: Yet,

text: surpris-

text: ﬁgure

text: shows

text: the

text: calculated

text: residual

text: norms

text: of

text: the

text: wave-

text: ingly,

text: the

text: band

text: gaps

text: vary

text: by

text: from

text: eV

text: in

text: function

text: of

text: after

text: projecting

text: it

text: to

text: the

text: s-

text: and

text: to

text: eV

text: in

text: despite

text: that

text: the

text: structures

text: are

text: locally

text: p-atomic

text: orbitals.

text: While

text: the

text: wavefunction

text: in

text: valence

text: identical

text: to

text: each

text: other

text: in

text: all

text: the

text: polytypes

text: This

text: phe-

text: bands

text: is

text: well

text: described

text: by

text: s

text: and

text: p

text: atomic

text: orbitals,

text: nomenon

text: was

text: diﬃcult

text: to

text: be

text: understood

text: in

text: conventional

text: conduction

text: electrons

text: cannot.

text: This

text: floating

text: character

text: at

text: chemical

text: pictures.

text: the

text: CBM

text: is

text: an

text: important

text: key

text: to

text: reveal

text: the

text: microscopic

text: We

text: have

text: recently

text: reported

text: the

text: microscopic

text: mech-

text: mechanism

text: of

text: the

text: band-gap

text: variations.

text: Channel

text: length

text: anism

text: of

text: band-gap

text: variations

text: in

text: SiC

text: polytypes

text: based

text: changes

text: depending

text: on

text: polytypes:

text: structure

text: has

text: in-

text: on

text: the

text: density

text: functional

text: theory

text: (DFT)

text: It

text: is

text: ﬁnite

text: channel

text: length

text: along

text: the

text: direction,

text: while

text: found

text: that

text: continuum

text: states

text: exist

text: in

text: conduction

text: bands

text: similar

text: channel

text: structure

text: is

text: seen

text: also

text: in

text: structure

text: of

text: materials,

text: and

text: furthermore,

text: such

text: peculiar

text: with

text: the

text: length

text: of

text: about

text: along

text: which

text: electron

text: state

text: appears

text: at

text: the

text: conduction-band

text: minima

text: is

text: slanted

text: relative

text: to

text: direction

text: with

text: a

text: a

text: lattice

text: (CBM)

text: in

text: SiC

text: polytypes.

text: The

text: wavefunction

text: at

text: the

text: CBM

text: constant.

text: Floating

text: states

text: at

text: the

text: CBM

text: extends

text: in

text: the

text: is

text: not

text: distributed

text: near

text: atomic

text: sites,

text: but

text: extends

text: (or

text: internal

text: space.

text: Therefore,

text: channel

text: length

text: and

text: channel

text: floats)

text: in

text: interstitial

text: channels

text: without

text: atomic

text: orbital

text: shapes

text: are

text: decisive

text: in

text: the

text: positions

text: of

text: the

text: CBM

text: in

text: energy

text: ##LTFigure##

text: ##LTLine##


text: 8 32, 783 (1979). [29] K. Kobayashi, and S. Komatsu, J. Phys. Soc. Jpn., 81, [21] R. Kaplan, R. J. Wagner, H. J. Kim, and R. F. Davis, 024714 (2012). Solid State Commun. 55, 67 (1985). [30] K. Kobayashi, and S. Komatsu, J. Phys. Soc. Jpn., 77, [22] N. T. Son, O. Kordina, A. O. Konstantinov, W. M. Chen, 084703 (2008). E. Sorman, B. Monemar, and E. Janzen, Appl. Phys. [31] M. S. Hybertsen and S. G. Louie, Phys. Rev. B 34, 5390 Lett. 65, 3209 (1994). (1986). [23] H. Schulz and K. H. Thiemann, Solid State Commun. [32] F. Aryasetiawan and O. Gunnarsson, Rep. Prog. Phys. 23, 815 (1977). 61, 237 (1998). [24] it Properties of Group III Nitrides, ed. J. H. Edgar (IEE, [33] For discussion on several ambiguities in the GW approxi- London, 1994) EMIS Datareviews Series. mation, see M. L. Tiago, S. Ismail-Beigi and S. G. Louie, [25] O. Mishima and K. Era: in it Electric Refractory Materi- Phys. Rev. B 69, 125212 (2004); M. van Schilfgaarde, T. als, ed. Y. Kumashiro (Marcel Dekker, New York, 2000) Kotani and S.V. Faleev, Phys. Rev. B 74, 245125 (2006), pp. 495-556, and references therein. and references therein. [26] S. Komatsu, K. Kobayashi, et. al., J. Phys. Chem. C, [34] H. Yamashita, K. Fukui, S. Misawa, and S. Yoshida, J. 114, 13176 (2010). Appl. Phys.50, 896 (1979). [27] H. P. Maruska, and J. J. Tietjen, Appl. Phys. Lett., 15, [35] P. T. Margarita, W. A. Gregory, S. Z. Tsvetanka, A. 327 (1969). J. Kenneth, J. S. Steven, and J. N. Hilﬁker, J. Appl. [28] I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Phys.89, 3331 (2001). Appl. Phys.89, 5815 (2001). [36] R. M. Chrenko, Solid State Comm., 14, 511 (1974).

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text: AKAMINE

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text: Cruceanu,

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text: Yamada,

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text: surfaces,

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text: surfaces,

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text: Koh,

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text: W.

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text: Yamada,

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text: bolic

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text: Math.

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text: Milnor,

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text: J.

text: Y.

text: Ogata

text: and

text: K.

text: Teramoto,

text: Duality

text: between

text: cuspidal

text: butterﬂies

text: and

text: cuspidal

text: S

text: singularities

text: on

text: maxfaces,

text: preprint.

text: Z.

text: Olszak,

text: A

text: note

text: about

text: the

text: torsion

text: of

text: curves

text: in

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text: Minkowski

text: spacetime

text: and

text: the

text: Schwarzian

text: derivative,

text: Filomat

text: K.

text: Saji,

text: M.

text: Umehara

text: and

text: K.

text: Yamada,

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text: geometry

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text: fronts,

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text: Math.

text: H.

text: Takahashi,

text: Timelike

text: minimal

text: surfaces

text: with

text: singularities

text: in

text: three-dimensional

text: spacetime

text: (in

text: Japan-

text: ese),

text: Master

text: thesis,

text: Osaka

text: University

text: M.

text: Umehara

text: and

text: K.

text: Yamada,

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text: surfaces

text: with

text: singularities

text: in

text: Minkowski

text: space,

text: Hokkaido

text: Math.

text: J.

text: E.

text: Vessiot,

text: Sur

text: les

text: curbes

text: minima,

text: C.

text: R.

text: Acad.

text: Sci.,

text: Paris

text: T.

text: Weinstein,

text: An

text: Introduction

text: to

text: Lorentz

text: Surfaces,

text: de

text: Gruyter

text: Exposition

text: in

text: Math.

text: Walter

text: de

text: Gruyter,

text: Berlin

text: Graduate

text: School

text: of

text: Mathematics

text: Kyushu

text: University,

text: Motooka,

text: Nishi-ku,

text: Fukuoka

text: Japan

text: Email

text: address:

text: s-akamine@math.kyushu-u.ac.jp