text: Page 2 of 15 P. V. M. BLAGOJEVIC´, F. FRICK & G. M. ZIEGLER Our machinery uses the topological Tverberg theorem as a black box: It provides the core for our proofs, but the proof technique relies only on the result, not on its proof. Thus, for example, an extension of the topological Tverberg theorem to some r that are not prime powers would immediately yield the same extension for our derived results. In place of the black box result we can also use the optimal colored Tverberg theorem; in Section we thus obtain further results of the “colored Tverberg” type. The black box result can also be replaced by Tverberg’s original theorem if the constraint functions are aﬃne; in this case no prime power restriction on the number of parts is needed. Thus we obtain a new aﬃne version of the topological Tverberg theorem for j-wise disjoint simplices; see Section . Similarly, from the Tverberg theorem for maps to manifolds by Volovikov [20] one “automatically” gets extensions of our results for maps to manifolds. 2. A brief history of Tverberg type results For every drawing of a tetrahedron in the plane either a vertex will end up on top of the opposite face or two opposite edges intersect in the drawing. This is a ﬁrst instance of a Tverberg type result, which holds true in any dimension as was proved by Johann Radon (1921): ⊆ R Radon’s theorem. For any subset X d with (at least) d + 2 elements there are ∩ (cid:54) ∅ disjoint subsets S and T of X with the property that conv(S) conv(T) = . Radon’s theorem has an equivalent reformulation in terms of aﬃne maps of the (d + 1)- R dimensional standard simplex ∆ to d: d+1 → R Affine Radon theorem. For every aﬃne map f : ∆ d there are disjoint faces ∩ (cid:54) d∅+1 σ and τ of ∆ with the property that f(σ) f(τ) = . d+1 Now we can see many possible ways to extend this basic result. First, we can ask whether the assumption on f to be an aﬃne map is essential. Is it enough to assume that f is only continuous? This question was answered by Bajm´oczy & B´ar´any [2] in 1979 via a clever use of Borsuk–Ulam theorem: → R Topological Radon theorem. For any continuous map f : ∆ d there are d+1 ∩ (cid:54) ∅ disjoint faces σ and τ of the simplex ∆ with the property that f(σ) f(τ) = . d+1 It is also natural to ask for more than two pairwise disjoint subsets of a suﬃciently large set ⊂ R of points X d. Such a result was ﬁrst achieved by Birch [6] in 1959 for d = 2, who also conjectured the tight result for d > 2, which was eventually proved by Helge Tverberg [19] in 1966. In the equivalent aﬃne version, his result reads as follows: ≥ ≥ − Affine Tverberg theorem. Let d 1 and r 2 be integers, and N = (r 1)(d + 1). → R For any aﬃne map f : ∆ d there are r pairwise disjoint faces σ , . . . , σ of ∆ such ∩ · · · ∩ N(cid:54) ∅ 1 r N that f(σ ) f(σ ) = . { 1 } r The set σ , . . . , σ of disjoint faces of ∆ whose images intersect is called a Tverberg partition 1→ R r N for f : ∆ d. The dimension of the simplex ∆ in the theorem is minimal. N N Will the statement of Tverberg’s theorem still be true if the map f is only assumed to be continuous? Surprisingly, the available answers to this question depend on divisibility properties of the parameter r. B´ar´any, Shlosman & Szu˝cs [5] in 1981 formulated the corresponding “topological Tverberg theorem” for continuous maps and proved it for the case r is a prime number. This was extended to all prime powers r by O¨zaydin [15] in 1987: ≥ ≥ − Topological Tverberg theorem. Let r 2, d 1, and N = (r 1)(d + 1). If r is → R a prime power, then for every continuous map f : ∆ d there are r pairwise disjoint ∩ · · · ∩ N(cid:54) ∅ faces σ , . . . , σ of ∆ such that f(σ ) f(σ ) = . 1 r N 1 r The topological Tverberg theorem is derived from the nonexistence of an S -equivariant ∗ ⊕ r map from the join [r] (N+1) into the sphere S(W (d+1)) when r is a prime power. Here { } r [r] = 1, . . . , r denotes a 0-dimension simplicial complex of r distinct points with the { · · · } obvious action of the symmetric group S , and W := (x , . . . , x ) : x + + x = 0 is r r 1 r 1 r an S -representation with an action given by permutation of coordinates. The fact that an r ∗ → ⊕ S -equivariant map [r] (N+1) S(W (d+1)) exists if r is not a prime power, established in [8], r r

text: Page

text: of

text: P.

text: V.

text: M.

text: BLAGOJEVIC´,

text: F.

text: FRICK

text: G.

text: M.

text: ZIEGLER

text: Our

text: machinery

text: uses

text: the

text: topological

text: Tverberg

text: theorem

text: as

text: a

text: black

text: box:

text: It

text: provides

text: the

text: core

text: for

text: our

text: proofs,

text: but

text: the

text: proof

text: technique

text: relies

text: only

text: on

text: the

text: result,

text: not

text: on

text: its

text: proof.

text: Thus,

text: for

text: example,

text: an

text: extension

text: of

text: the

text: topological

text: Tverberg

text: theorem

text: to

text: some

text: r

text: that

text: are

text: not

text: prime

text: powers

text: would

text: immediately

text: yield

text: the

text: same

text: extension

text: for

text: our

text: derived

text: results.

text: In

text: place

text: of

text: the

text: black

text: box

text: result

text: we

text: can

text: also

text: use

text: the

text: optimal

text: colored

text: Tverberg

text: theorem;

text: in

text: Section

text: we

text: thus

text: obtain

text: further

text: results

text: of

text: the

text: “colored

text: Tverberg”

text: type.

text: The

text: black

text: box

text: result

text: can

text: also

text: be

text: replaced

text: by

text: Tverberg’s

text: original

text: theorem

text: if

text: the

text: constraint

text: functions

text: are

text: aﬃne;

text: in

text: this

text: case

text: no

text: prime

text: power

text: restriction

text: on

text: the

text: number

text: of

text: parts

text: is

text: needed.

text: Thus

text: we

text: obtain

text: a

text: new

text: aﬃne

text: version

text: of

text: the

text: topological

text: Tverberg

text: theorem

text: for

text: j-wise

text: disjoint

text: simplices;

text: see

text: Section

text: Similarly,

text: from

text: the

text: Tverberg

text: theorem

text: for

text: maps

text: to

text: manifolds

text: by

text: Volovikov

text: one

text: “automatically”

text: gets

text: extensions

text: of

text: our

text: results

text: for

text: maps

text: to

text: manifolds.

text: A

text: brief

text: history

text: of

text: Tverberg

text: type

text: results

text: For

text: every

text: drawing

text: of

text: a

text: tetrahedron

text: in

text: the

text: plane

text: either

text: a

text: vertex

text: will

text: end

text: up

text: on

text: top

text: of

text: the

text: opposite

text: face

text: or

text: two

text: opposite

text: edges

text: intersect

text: in

text: the

text: drawing.

text: This

text: is

text: a

text: ﬁrst

text: instance

text: of

text: a

text: Tverberg

text: type

text: result,

text: which

text: holds

text: true

text: in

text: any

text: dimension

text: as

text: was

text: proved

text: by

text: Johann

text: Radon

text: R

text: Radon’s

text: theorem.

text: For

text: any

text: subset

text: X

text: d

text: with

text: (at

text: least)

text: d

text: elements

text: there

text: are

text: disjoint

text: subsets

text: S

text: and

text: T

text: of

text: X

text: with

text: the

text: property

text: that

text: conv(S)

text: conv(T)

text: Radon’s

text: theorem

text: has

text: an

text: equivalent

text: reformulation

text: in

text: terms

text: of

text: aﬃne

text: maps

text: of

text: the

text: (d

text: R

text: dimensional

text: standard

text: simplex

text: to

text: d:

text: R

text: Affine

text: Radon

text: theorem.

text: For

text: every

text: aﬃne

text: map

text: f

text: d

text: there

text: are

text: disjoint

text: faces

text: σ

text: and

text: τ

text: of

text: with

text: the

text: property

text: that

text: f(σ)

text: f(τ)

text: Now

text: we

text: can

text: see

text: many

text: possible

text: ways

text: to

text: extend

text: this

text: basic

text: result.

text: First,

text: we

text: can

text: ask

text: whether

text: the

text: assumption

text: on

text: f

text: to

text: be

text: an

text: aﬃne

text: map

text: is

text: essential.

text: Is

text: it

text: enough

text: to

text: assume

text: that

text: f

text: is

text: only

text: continuous?

text: This

text: question

text: was

text: answered

text: by

text: Bajm´oczy

text: B´ar´any

text: in

text: via

text: a

text: clever

text: use

text: of

text: Borsuk–Ulam

text: theorem:

text: R

text: Topological

text: Radon

text: theorem.

text: For

text: any

text: continuous

text: map

text: f

text: d

text: there

text: are

text: disjoint

text: faces

text: σ

text: and

text: τ

text: of

text: the

text: simplex

text: with

text: the

text: property

text: that

text: f(σ)

text: f(τ)

text: It

text: is

text: also

text: natural

text: to

text: ask

text: for

text: more

text: than

text: two

text: pairwise

text: disjoint

text: subsets

text: of

text: a

text: suﬃciently

text: large

text: set

text: R

text: of

text: points

text: X

text: d.

text: Such

text: a

text: result

text: was

text: ﬁrst

text: achieved

text: by

text: Birch

text: in

text: for

text: d

text: who

text: also

text: conjectured

text: the

text: tight

text: result

text: for

text: d

text: which

text: was

text: eventually

text: proved

text: by

text: Helge

text: Tverberg

text: in

text: In

text: the

text: equivalent

text: aﬃne

text: version,

text: his

text: result

text: reads

text: as

text: follows:

text: Affine

text: Tverberg

text: theorem.

text: Let

text: d

text: and

text: r

text: be

text: integers,

text: and

text: N

text: (r

text: R

text: For

text: any

text: aﬃne

text: map

text: f

text: d

text: there

text: are

text: r

text: pairwise

text: disjoint

text: faces

text: σ

text: σ

text: of

text: such

text: r

text: N

text: that

text: f(σ

text: f(σ

text: r

text: The

text: set

text: σ

text: σ

text: of

text: disjoint

text: faces

text: of

text: whose

text: images

text: intersect

text: is

text: called

text: a

text: Tverberg

text: partition

text: 1→

text: R

text: r

text: N

text: for

text: f

text: d.

text: The

text: dimension

text: of

text: the

text: simplex

text: in

text: the

text: theorem

text: is

text: minimal.

text: N

text: N

text: Will

text: the

text: statement

text: of

text: Tverberg’s

text: theorem

text: still

text: be

text: true

text: if

text: the

text: map

text: f

text: is

text: only

text: assumed

text: to

text: be

text: continuous?

text: Surprisingly,

text: the

text: available

text: answers

text: to

text: this

text: question

text: depend

text: on

text: divisibility

text: properties

text: of

text: the

text: parameter

text: r.

text: B´ar´any,

text: Shlosman

text: Szu˝cs

text: in

text: formulated

text: the

text: corresponding

text: “topological

text: Tverberg

text: theorem”

text: for

text: continuous

text: maps

text: and

text: proved

text: it

text: for

text: the

text: case

text: r

text: is

text: a

text: prime

text: number.

text: This

text: was

text: extended

text: to

text: all

text: prime

text: powers

text: r

text: by

text: O¨zaydin

text: in

text: Topological

text: Tverberg

text: theorem.

text: Let

text: r

text: d

text: and

text: N

text: (r

text: If

text: r

text: is

text: R

text: a

text: prime

text: power,

text: then

text: for

text: every

text: continuous

text: map

text: f

text: d

text: there

text: are

text: r

text: pairwise

text: disjoint

text: faces

text: σ

text: σ

text: of

text: such

text: that

text: f(σ

text: f(σ

text: r

text: N

text: r

text: The

text: topological

text: Tverberg

text: theorem

text: is

text: derived

text: from

text: the

text: nonexistence

text: of

text: an

text: S

text: -equivariant

text: r

text: map

text: from

text: the

text: join

text: [r]

text: into

text: the

text: sphere

text: S(W

text: when

text: r

text: is

text: a

text: prime

text: power.

text: Here

text: r

text: [r]

text: r

text: denotes

text: a

text: simplicial

text: complex

text: of

text: r

text: distinct

text: points

text: with

text: the

text: obvious

text: action

text: of

text: the

text: symmetric

text: group

text: S

text: and

text: W

text: (x

text: x

text: x

text: x

text: is

text: r

text: r

text: r

text: r

text: an

text: S

text: -representation

text: with

text: an

text: action

text: given

text: by

text: permutation

text: of

text: coordinates.

text: The

text: fact

text: that

text: an

text: r

text: S

text: -equivariant

text: map

text: [r]

text: S(W

text: exists

text: if

text: r

text: is

text: not

text: a

text: prime

text: power,

text: established

text: in

text: r

text: r


text: TVERBERG PLUS CONSTRAINTS Page 3 of 15 makes the extension of topological Tverberg theorem to non prime powers into “one of the most challenging problems in this ﬁeld” [14, Notes to Sect. 6.4]. What is next? Can we say something about dimensions of simplexes in a Tverberg partition? What is the minimal dimension of a simplex ∆ such that for every mapping to the plane there are two disjoint edges whose images intersect? (The answer to this is given by Kuratowski’s theorem from graph theory: K is not planar, so the 4-simplex will do.) How about higher- 5 dimensional versions of this result? The classical van Kampen–Flores theorem [10] [12] from the 1930s provides a ﬁrst answer to such questions. ≥ → R van Kampen–Flores theorem. For d 2 even, and a continuous map f : ∆ d d+2 there are two disjoint faces σ , σ of ∆ of dimension at most d in ∆ such that ∩ (cid:54) ∅ 1 2 d+2 2 d+2 f(σ ) f(σ ) = . 1 2 The generalized van Kampen–Flores theorems of Sarkaria [17] and Volovikov [21] extend this by providing conditions that guarantee the existence of multiple overlaps. These theorems are classically obtained via topological methods that are considerably more involved than those needed for to the proof of topological Tverberg theorem in cases of primes or of prime powers. Can we put any other restrictions on a Tverberg partition? B´ar´any, Fu¨redi & Lov´asz [3] in 1989 realized a need for “a colored version of Tverberg’s theorem.” They proved the ﬁrst instance of a such a result for three triangles in the plane. Extending these ideas, B´ar´any & Larman [4] in 1992 formulated the following general problem. The colored Tverberg problem. Determine the smallest natural number n = n(d, r) C (cid:93) · · · (cid:93) R such that for every collection = C C of n points in d, where each “color class” | | ≥ 0 d C C satisﬁes C r, there are r disjoint subcollections F , . . . , F of such that i | ∩ i | ≤ ∈ { } ∈ { 1} r (A) C F 1 for every i 0, . . . , d , j 1, . . . , r , and i j ∩ · · · ∩ (cid:54) ∅ (B) conv (F ) conv (F ) = 1 r They proved that n(1, r) = 2r, n(2, r) = 3r, presented a proof by Lov´asz for n(d, 2) = 2(d + 1), and conjectured that n(d, r) = r(d + 1). In the same year Zˇivaljevi´c & Vre´cica [25] formulated the following modiﬁed colored Tverberg problem: Modified colored Tverberg problem. Determine the smallest number t = tt(d, r) C (cid:93) · · · (cid:93) such that for every simplex ∆ with (d + 1)-colored vertex set = C C , with | | ≥ → R 0 d C t for all i, and for every continuous map f : ∆ d, there are r disjoint faces i σ , . . . , σ of ∆ satisfying 1 | r ∩ | ≤ ∈ { } ∈ { } (A) C σ 1 for every i 0, . . . , d , j 1, . . . , r , and i ∩j · · · ∩ (cid:54) ∅ (B) f(σ ) f(σ ) = . 1 r ≤ − Zˇivaljevi´c and Vre´cica obtained the upper bound for the function tt(d, r) 2r 1 in the case ≤ − when r is a prime that yield upper bound tt(d, r) 4r 3 for any r. (Zˇivaljevi´c later extended this to prime powers r.) These results were obtained by proving the nonexistence of an S - ∗ ⊕ r equivariant map from the join ∆ (d+−1) into the sphere S(W (d+1)) when r is a prime power. × r−,2r 1 r Here ∆ − denotes the r (2r 1) chessboard complex. The symmetric group S acts on r,2r 1 r ∆ − by permuting the rows. The connectivity of the chessboard ∆ − plays the central r,2r 1 ∗ r,2→r 1 ⊕ role in the proof of the nonexistence of an S -equivariant map ∆ (d+−1) S(W (d+1)). Note, r r,2r 1 r however, that the upper bound of Zˇivaljevi´c and Vre´cica on the function tt(d, r) does not provide any information about the function n(d, r) of B´ar´any and Larman. Only recently Blagojevi´c, Matschke & Ziegler [7], as a consequence of their “optimal colored ≤ Tverberg theorem”, established that n(d, r) = r(d + 1) when r + 1 is a prime and n(d, r) − ≤ − 2(r 1)(d + 1) + 1 when r is a prime. Furthermore, they established tt(d, r) 2r 2 for any r. See [24] for an exposition. Can we ask for even more? What about preimages of the intersection point of a Tverberg partition? Can we say something about the barycentric coordinates of these preimages? This question was addressed by Sober´on [18] in 2013. This result is discussed in more detail in Section . ##LTLine##

text: TVERBERG

label: paragraph

text: PLUS

text: CONSTRAINTS

text: Page

text: of

text: makes

text: the

text: extension

text: of

text: topological

text: Tverberg

text: theorem

text: to

text: non

text: prime

text: powers

text: into

text: “one

text: of

text: the

text: most

text: challenging

text: problems

text: in

text: this

text: ﬁeld”

text: Notes

text: to

text: Sect.

text: What

text: is

text: next?

text: Can

text: we

text: say

text: something

text: about

text: dimensions

text: of

text: simplexes

text: in

text: a

text: Tverberg

text: partition?

text: What

text: is

text: the

text: minimal

text: dimension

text: of

text: a

text: simplex

text: such

text: that

text: for

text: every

text: mapping

text: to

text: the

text: plane

text: there

text: are

text: two

text: disjoint

text: edges

text: whose

text: images

text: intersect?

text: (The

text: answer

text: to

text: this

text: is

text: given

text: by

text: Kuratowski’s

text: theorem

text: from

text: graph

text: theory:

text: K

text: is

text: not

text: planar,

text: so

text: the

text: will

text: do.)

text: How

text: about

text: higher-

text: dimensional

text: versions

text: of

text: this

text: result?

text: The

text: classical

text: van

text: Kampen–Flores

text: theorem

text: from

text: the

text: provides

text: a

text: ﬁrst

text: answer

text: to

text: such

text: questions.

text: R

text: van

text: Kampen–Flores

text: theorem.

text: For

text: d

text: even,

text: and

text: a

text: continuous

text: map

text: f

text: d

text: there

text: are

text: two

text: disjoint

text: faces

text: σ

text: σ

text: of

text: of

text: dimension

text: at

text: most

text: d

text: in

text: such

text: that

text: f(σ

text: f(σ

text: The

text: generalized

text: van

text: Kampen–Flores

text: theorems

text: of

text: Sarkaria

text: and

text: Volovikov

text: extend

text: this

text: by

text: providing

text: conditions

text: that

text: guarantee

text: the

text: existence

text: of

text: multiple

text: overlaps.

text: These

text: theorems

text: are

text: classically

text: obtained

text: via

text: topological

text: methods

text: that

text: are

text: considerably

text: more

text: involved

text: than

text: those

text: needed

text: for

text: to

text: the

text: proof

text: of

text: topological

text: Tverberg

text: theorem

text: in

text: cases

text: of

text: primes

text: or

text: of

text: prime

text: powers.

text: Can

text: we

text: put

text: any

text: other

text: restrictions

text: on

text: a

text: Tverberg

text: partition?

text: B´ar´any,

text: Fu¨redi

text: Lov´asz

text: in

text: realized

text: a

text: need

text: for

text: “a

text: colored

text: version

text: of

text: Tverberg’s

text: theorem.”

text: They

text: proved

text: the

text: ﬁrst

text: instance

text: of

text: a

text: such

text: a

text: result

text: for

text: three

text: triangles

text: in

text: the

text: plane.

text: Extending

text: these

text: ideas,

text: B´ar´any

text: Larman

text: in

text: formulated

text: the

text: following

text: general

text: problem.

text: The

text: colored

text: Tverberg

text: problem.

text: Determine

text: the

text: smallest

text: natural

text: number

text: n

text: n(d,

text: r)

text: C

text: R

text: such

text: that

text: for

text: every

text: collection

text: C

text: C

text: of

text: n

text: points

text: in

text: d,

text: where

text: each

text: “color

text: class”

text: d

text: C

text: C

text: satisﬁes

text: C

text: r,

text: there

text: are

text: r

text: disjoint

text: subcollections

text: F

text: F

text: of

text: such

text: that

text: i

text: i

text: r

text: (A)

text: C

text: F

text: for

text: every

text: i

text: d

text: j

text: r

text: and

text: i

text: j

text: (B)

text: conv

text: (F

text: conv

text: (F

text: r

text: They

text: proved

text: that

text: r)

text: r)

text: presented

text: a

text: proof

text: by

text: Lov´asz

text: for

text: n(d,

text: and

text: conjectured

text: that

text: n(d,

text: r)

text: r(d

text: In

text: the

text: same

text: year

text: Zˇivaljevi´c

text: Vre´cica

text: formulated

text: the

text: following

text: modiﬁed

text: colored

text: Tverberg

text: problem:

text: Modified

text: colored

text: Tverberg

text: problem.

text: Determine

text: the

text: smallest

text: number

text: t

text: tt(d,

text: r)

text: C

text: such

text: that

text: for

text: every

text: simplex

text: with

text: (d

text: vertex

text: set

text: C

text: C

text: with

text: R

text: d

text: C

text: t

text: for

text: all

text: i,

text: and

text: for

text: every

text: continuous

text: map

text: f

text: d,

text: there

text: are

text: r

text: disjoint

text: faces

text: i

text: σ

text: σ

text: of

text: satisfying

text: r

text: (A)

text: C

text: σ

text: for

text: every

text: i

text: d

text: j

text: r

text: and

text: i

text: ∩j

text: (B)

text: f(σ

text: f(σ

text: r

text: Zˇivaljevi´c

text: and

text: Vre´cica

text: obtained

text: the

text: upper

text: bound

text: for

text: the

text: function

text: tt(d,

text: r)

text: in

text: the

text: case

text: when

text: r

text: is

text: a

text: prime

text: that

text: yield

text: upper

text: bound

text: tt(d,

text: r)

text: for

text: any

text: r.

text: (Zˇivaljevi´c

text: later

text: extended

text: this

text: to

text: prime

text: powers

text: r.)

text: These

text: results

text: were

text: obtained

text: by

text: proving

text: the

text: nonexistence

text: of

text: an

text: S

text: r

text: equivariant

text: map

text: from

text: the

text: join

text: into

text: the

text: sphere

text: S(W

text: when

text: r

text: is

text: a

text: prime

text: power.

text: r

text: Here

text: denotes

text: the

text: r

text: chessboard

text: complex.

text: The

text: symmetric

text: group

text: S

text: acts

text: on

text: r

text: by

text: permuting

text: the

text: rows.

text: The

text: connectivity

text: of

text: the

text: chessboard

text: plays

text: the

text: central

text: role

text: in

text: the

text: proof

text: of

text: the

text: nonexistence

text: of

text: an

text: S

text: -equivariant

text: map

text: S(W

text: Note,

text: r

text: r

text: however,

text: that

text: the

text: upper

text: bound

text: of

text: Zˇivaljevi´c

text: and

text: Vre´cica

text: on

text: the

text: function

text: tt(d,

text: r)

text: does

text: not

text: provide

text: any

text: information

text: about

text: the

text: function

text: n(d,

text: r)

text: of

text: B´ar´any

text: and

text: Larman.

text: Only

text: recently

text: Blagojevi´c,

text: Matschke

text: Ziegler

text: as

text: a

text: consequence

text: of

text: their

text: “optimal

text: colored

text: Tverberg

text: theorem”,

text: established

text: that

text: n(d,

text: r)

text: r(d

text: when

text: r

text: is

text: a

text: prime

text: and

text: n(d,

text: r)

text: when

text: r

text: is

text: a

text: prime.

text: Furthermore,

text: they

text: established

text: tt(d,

text: r)

text: for

text: any

text: r.

text: See

text: for

text: an

text: exposition.

text: Can

text: we

text: ask

text: for

text: even

text: more?

text: What

text: about

text: preimages

text: of

text: the

text: intersection

text: point

text: of

text: a

text: Tverberg

text: partition?

text: Can

text: we

text: say

text: something

text: about

text: the

text: barycentric

text: coordinates

text: of

text: these

text: preimages?

text: This

text: question

text: was

text: addressed

text: by

text: Sober´on

text: in

text: This

text: result

text: is

text: discussed

text: in

text: more

text: detail

text: in

text: Section

text: ##LTLine##


text: FIG. 4: (Color online) PDD distributions in an inhomogeneous phantom for a beam of 10 mm cone size. The solid and the dashed lines represent dose measured with EBT ﬁlm and that calculated without the EBT ﬁlm, respectively [? ]. The open circles and the open squares denote our calculated doses with and without EBT ﬁlm, respectively. GEANT4 DTP to the GEANT4 DTF. Second, we compare our calculated PDD with the calculated and measured PDD [? ] for inhomogeneous phantoms. Figure ?? (a) and (b) show the PDD in inhomogeneous phantom with the Balsa wood and cortical bone, respectively. In the case when the Balsa wood is inserted in the phantom, our calculations reproduce well the PDD measured and calculated in Ref. [? ]. Figure ?? (a) shows some diﬀerence between the doses with and without EBT ﬁlms in our calculations as observed in the previous results [? ] in the region beyond the Balsa wood. Also, Fig. ?? (b) shows that when cortical bone equivalent material is inserted in the phantom our calculations with and without EBT ﬁlm reproduce the PDD measured with EBT ﬁlm [? ] and the PDD calculated without EBT ﬁlm, respectively. Thus previous calculations and measurements [? ] are well reproduced by our calculations, which conﬁrms the accuracy of our GEANT4 calculations for both cases with and without the EBT ﬁlm. This results also shows that the presence of the EBT ﬁlm aﬀects the value of PDD for a small cone size of 5 mm and needs to be properly taken into consideration in the simulation. B. Comparison of simulations with experiments with inhomogeneity of water- lung-water We performed the calculations and measurements of PDDs for an inhomogeneous phan- tom with Balsa wood for three ﬁeld sizes of 5, 10, 30 mm, and the results are represented 11 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

label: caption

text: online)

text: PDD

text: distributions

text: in

text: an

text: inhomogeneous

text: phantom

text: for

text: a

text: beam

text: of

text: mm

text: cone

text: size.

text: The

text: solid

text: and

text: the

text: dashed

text: lines

text: represent

text: dose

text: measured

text: with

text: EBT

text: ﬁlm

text: and

text: that

text: calculated

text: without

text: the

text: EBT

text: ﬁlm,

text: respectively

text: The

text: open

text: circles

text: and

text: the

text: open

text: squares

text: denote

text: our

text: calculated

text: doses

text: with

text: and

text: without

text: EBT

text: ﬁlm,

text: respectively.

text: DTP

text: to

text: the

text: DTF.

text: Second,

text: we

text: compare

text: our

text: calculated

text: PDD

text: with

text: the

text: calculated

text: and

text: measured

text: PDD

text: for

text: inhomogeneous

text: phantoms.

text: Figure

text: (a)

text: and

text: (b)

text: show

text: the

text: PDD

text: in

text: inhomogeneous

text: phantom

text: with

text: the

text: Balsa

text: wood

text: and

text: cortical

text: bone,

text: respectively.

text: In

text: the

text: case

text: when

text: the

text: Balsa

text: wood

text: is

text: inserted

text: in

text: the

text: phantom,

text: our

text: calculations

text: reproduce

text: well

text: the

text: PDD

text: measured

text: and

text: calculated

text: in

text: Ref.

text: Figure

text: (a)

text: shows

text: some

text: diﬀerence

text: between

text: the

text: doses

text: with

text: and

text: without

text: EBT

text: ﬁlms

text: in

text: our

text: calculations

text: as

text: observed

text: in

text: the

text: previous

text: results

text: in

text: the

text: region

text: beyond

text: the

text: Balsa

text: wood.

text: Also,

text: Fig.

text: (b)

text: shows

text: that

text: when

text: cortical

text: bone

text: equivalent

text: material

text: is

text: inserted

text: in

text: the

text: phantom

text: our

text: calculations

text: with

text: and

text: without

text: EBT

text: ﬁlm

text: reproduce

text: the

text: PDD

text: measured

text: with

text: EBT

text: ﬁlm

text: and

text: the

text: PDD

text: calculated

text: without

text: EBT

text: ﬁlm,

text: respectively.

text: Thus

text: previous

text: calculations

text: and

text: measurements

text: are

text: well

text: reproduced

text: by

text: our

text: calculations,

text: which

text: conﬁrms

text: the

text: accuracy

text: of

text: our

text: calculations

text: for

text: both

text: cases

text: with

text: and

text: without

text: the

text: EBT

text: ﬁlm.

text: This

text: results

text: also

text: shows

text: that

text: the

text: presence

text: of

text: the

text: EBT

text: ﬁlm

text: aﬀects

text: the

text: value

text: of

text: PDD

text: for

text: a

text: small

text: cone

text: size

text: of

text: mm

text: and

text: needs

text: to

text: be

text: properly

text: taken

text: into

text: consideration

text: in

text: the

text: simulation.

text: B.

text: Comparison

text: of

text: simulations

text: with

text: experiments

text: with

text: inhomogeneity

text: of

text: water-

text: lung-water

text: We

text: performed

text: the

text: calculations

text: and

text: measurements

text: of

text: PDDs

text: for

text: an

text: inhomogeneous

text: phan-

text: tom

text: with

text: Balsa

text: wood

text: for

text: three

text: ﬁeld

text: sizes

text: of

text: mm,

text: and

text: the

text: results

text: are

text: represented

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: FIG. 5: (Color online) PDD distributions in an inhomogeneous phantom with Balsa wood for three ﬁeld sizes (5, 10 and 30 mm). The solid line denotes the PDD measured with the EBT ﬁlm. The dashed line represents the calculated dose values by using Multiplan. The open circles and the open squares denote the calculated PDD with and without the EBT ﬁlm, respectively. GEANT4 DTF reproduces well EXP DTF. in Fig. ??. The PDD calculated with the EBT ﬁlm (GEANT4 DTF) reproduce well the PDD measured with the EBT ﬁlm (EXP DTF). The average percentage errors between the GEANT4 DTF and the EXP DTF for 5, 10 and 30 mm of cone sizes are 2.20%, 0.43% and 0.08%, respectively. However, Multiplan cannot reproduce the measured dose values in both the Balsa wood region and beyond the Balsa wood region. Large discrepancies are observed especially in the Balsa wood region. The average ratio of the PDD obtained from Multiplan to those measured with EBT ﬁlm for 5, 10 and 30 mm of ﬁeld sizes are 2.1, 1.8 and 1.2, respectively, which shows larger discrepancies for smaller cone sizes. The same tendency can also be seen in Ref. [? ]. Figure ?? also shows there is a signiﬁcant diﬀerence between the PDD calculated without the EBT ﬁlm (GEANT4 DTP) and the PDD measured with the EBT ﬁlm. This clearly shows that the proper inclusion of the EBT ﬁlm in the simulation is necessary for accurate calculations of the dose. In Fig. ?? we have compared the GEANT4 DTP with the calculated and measured DTF. In the Balsa wood region, the average ratios 12 ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

text: online)

text: PDD

text: distributions

text: in

text: an

text: inhomogeneous

text: phantom

text: with

text: Balsa

text: wood

text: for

text: three

text: ﬁeld

text: sizes

text: and

text: mm).

text: The

text: solid

text: line

text: denotes

text: the

text: PDD

text: measured

text: with

text: the

text: EBT

text: ﬁlm.

text: The

text: dashed

text: line

text: represents

text: the

text: calculated

text: dose

text: values

text: by

text: using

text: Multiplan.

text: The

text: open

text: circles

text: and

text: the

text: open

text: squares

text: denote

text: the

text: calculated

text: PDD

text: with

text: and

text: without

text: the

text: EBT

text: ﬁlm,

text: respectively.

text: DTF

text: reproduces

text: well

text: EXP

text: DTF.

text: in

text: Fig.

text: The

text: PDD

text: calculated

text: with

text: the

text: EBT

text: ﬁlm

text: DTF)

text: reproduce

text: well

text: the

text: PDD

text: measured

text: with

text: the

text: EBT

text: ﬁlm

text: (EXP

text: DTF).

text: The

text: average

text: percentage

text: errors

text: between

text: the

text: DTF

text: and

text: the

text: EXP

text: DTF

text: for

text: and

text: mm

text: of

text: cone

text: sizes

text: are

text: and

text: respectively.

text: However,

text: Multiplan

text: cannot

text: reproduce

text: the

text: measured

text: dose

text: values

text: in

text: both

text: the

text: Balsa

text: wood

text: region

text: and

text: beyond

text: the

text: Balsa

text: wood

text: region.

text: Large

text: discrepancies

text: are

text: observed

text: especially

text: in

text: the

text: Balsa

text: wood

text: region.

text: The

text: average

text: ratio

text: of

text: the

text: PDD

text: obtained

text: from

text: Multiplan

text: to

text: those

text: measured

text: with

text: EBT

text: ﬁlm

text: for

text: and

text: mm

text: of

text: ﬁeld

text: sizes

text: are

text: and

text: respectively,

text: which

text: shows

text: larger

text: discrepancies

text: for

text: smaller

text: cone

text: sizes.

text: The

text: same

text: tendency

text: can

text: also

text: be

text: seen

text: in

text: Ref.

text: Figure

text: also

text: shows

text: there

text: is

text: a

text: signiﬁcant

text: diﬀerence

text: between

text: the

text: PDD

text: calculated

text: without

text: the

text: EBT

text: ﬁlm

text: DTP)

text: and

text: the

text: PDD

text: measured

text: with

text: the

text: EBT

text: ﬁlm.

text: This

text: clearly

text: shows

text: that

text: the

text: proper

text: inclusion

text: of

text: the

text: EBT

text: ﬁlm

text: in

text: the

text: simulation

text: is

text: necessary

text: for

text: accurate

text: calculations

text: of

text: the

text: dose.

text: In

text: Fig.

text: we

text: have

text: compared

text: the

text: DTP

text: with

text: the

text: calculated

text: and

text: measured

text: DTF.

text: In

text: the

text: Balsa

text: wood

text: region,

text: the

text: average

text: ratios

text: ##LTFigure##

text: ##LTFigure##


text: FIG. 8: (Color online) The ratios of the GEANT4 DTF and measured EXP DTF to the GEANT4 DTP are represented. The ratios of the calculated DTP obtained from Multiplan to the GEANT4 DTP are also shown. that the ratios of the DTF to DTP are nearly constant as 1.05 and independent of the cone sizes. Diﬀerence between the doses to water and the doses to other media was studied by Siebers et al. [? ]. They suggested correction factors for various beam energies and materials. For the case of the 6MV photon beams, correction factors needed for the lung, ICRU tissue, soft bone and cortical bone are 1, 1.101, 1.035 and 1.117, respectively. Wilcox et al. [? ] showed that the values of the DTP multiplied by the correction factor 1.117 in the cortical bone region agree with those of the doses obtained with the EBT ﬁlm. From our study, the ± ± average ratios of DTF to DTP are obtained to be 1.049 0.015 and 1.129 0.015 for soft bone and cortical bone, respectively. Our ratios (1.049 and 1.129) and the correction factors (1.035 and 1.117) of Ref. [? ] agree with each other within 1.3%. IV. CONCLUSION We have studied the PDD distributions in an inhomogeneous phantom with low and high density materials. Measurements were performed with Gafchromic EBT ﬁlm using the 6MV 15 ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

text: online)

text: The

text: ratios

text: of

text: the

text: DTF

text: and

text: measured

text: EXP

text: DTF

text: to

text: the

text: DTP

text: are

text: represented.

text: The

text: ratios

text: of

text: the

text: calculated

text: DTP

text: obtained

text: from

text: Multiplan

text: to

text: the

text: DTP

text: are

text: also

text: shown.

text: that

text: the

text: ratios

text: of

text: the

text: DTF

text: to

text: DTP

text: are

text: nearly

text: constant

text: as

text: and

text: independent

text: of

text: the

text: cone

text: sizes.

text: Diﬀerence

text: between

text: the

text: doses

text: to

text: water

text: and

text: the

text: doses

text: to

text: other

text: media

text: was

text: studied

text: by

text: Siebers

text: et

text: al.

text: They

text: suggested

text: correction

text: factors

text: for

text: various

text: beam

text: energies

text: and

text: materials.

text: For

text: the

text: case

text: of

text: the

text: photon

text: beams,

text: correction

text: factors

text: needed

text: for

text: the

text: lung,

text: ICRU

text: tissue,

text: soft

text: bone

text: and

text: cortical

text: bone

text: are

text: and

text: respectively.

text: Wilcox

text: et

text: al.

text: showed

text: that

text: the

text: values

text: of

text: the

text: DTP

text: multiplied

text: by

text: the

text: correction

text: factor

text: in

text: the

text: cortical

text: bone

text: region

text: agree

text: with

text: those

text: of

text: the

text: doses

text: obtained

text: with

text: the

text: EBT

text: ﬁlm.

text: From

text: our

text: study,

text: the

text: average

text: ratios

text: of

text: DTF

text: to

text: DTP

text: are

text: obtained

text: to

text: be

text: and

text: for

text: soft

text: bone

text: and

text: cortical

text: bone,

text: respectively.

text: Our

text: ratios

text: and

text: and

text: the

text: correction

text: factors

text: and

text: of

text: Ref.

text: agree

text: with

text: each

text: other

text: within

text: IV.

text: CONCLUSION

text: We

text: have

text: studied

text: the

text: PDD

text: distributions

text: in

text: an

text: inhomogeneous

text: phantom

text: with

text: low

text: and

text: high

text: density

text: materials.

text: Measurements

text: were

text: performed

text: with

text: Gafchromic

text: EBT

text: ﬁlm

text: using

text: the

text: ##LTFigure##

text: ##LTFigure##


text: accurate dose estimation is needed for tissues with inhomogeneous region, such as air cavity, lung and bones. Agreement between the treatment plan and the measured dose values for small ﬁeld radiations in inhomogeneous region in spine and lung tissues is a major factor that aﬀects treatment accuracy. Techniques to correct for inhomogeneity have been developed with increasing computing power and improved understanding of density-variable regions [? ]. Accurate dose measurement is particularly important in the dosimetry of small ﬁelds in the presence of low density inhomogeneities. When the photon beams pass through the air cavity, which is the lowest density area in the human body, the dose drops in the air cavity, whereas it gets built-up in the phantom, and thus a dose enhancement may be developed in the distal side of cavity [? ? ? ? ]. The characteristics of the inhomogeneous conditions were studied with the radiochromic ﬁlm type MD-55 (Gafchromic type MD-55 Cat no 37-041 Lot no J1548MD55) by Paelinck et al. [? ? ]. They performed the dose measurement with ﬁlms and Monte Carlo simulations for inhomogeneous phantoms involving the air cavity or lung equivalent media with the photon beams provided by 6MV linear accelerator. In their simulations, they modeled the phantoms and the ﬁlm by using BEAMnrc/EGSnrc system and then compared the simulation results with measurements, obtaining consistency between the ﬁlm measurements and the calculations with including the ﬁlm. The dose calculations were performed in two diﬀerent methods. First, the simulations were performed with a ﬁlm detector, and thus the doses were obtained from the ﬁlm. We refer to these values of dose as the ”dose to ﬁlm (DTF)”. Second, the simulations were done with phantom geometry without including the ﬁlm. In this case, the absorbed doses were calculated in the phantom material itself without taking into account the presence of the ﬁlm. So, we call these values of dose the ”dose to phantom (DTP)”. For the case when the air cavity was inserted in the phantom, they found a diﬀerence between the DTF and the DTP [? ? ]. They showed that the diﬀerences decrease as the ﬁeld size and the density of the medium increase. Also they found that these diﬀerences disappear by oﬀsetting the ﬁlm sideway by a few centimeters. A study of PDD for inhomogeneous phantoms with a Balsa wood as a low density material or a cortical bone as a high density material was done by Wilcox and Daskalov [? ], who compared the EBT ﬁlm measurements of PDD (DTF) with the calculations of DTP for inhomogeneous conditions by using Cyberknife systems. In that work, they did not include the presence of the EBT ﬁlm in the simulation since the ﬁlm thickness was smaller than the 4

text: accurate

text: dose

text: estimation

text: is

text: needed

text: for

text: tissues

text: with

text: inhomogeneous

text: region,

text: such

text: as

text: air

text: cavity,

text: lung

text: and

text: bones.

text: Agreement

text: between

text: the

text: treatment

text: plan

text: and

text: the

text: measured

text: dose

text: values

text: for

text: small

text: ﬁeld

text: radiations

text: in

text: inhomogeneous

text: region

text: in

text: spine

text: and

text: lung

text: tissues

text: is

text: a

text: major

text: factor

text: that

text: aﬀects

text: treatment

text: accuracy.

text: Techniques

text: to

text: correct

text: for

text: inhomogeneity

text: have

text: been

text: developed

text: with

text: increasing

text: computing

text: power

text: and

text: improved

text: understanding

text: of

text: density-variable

text: regions

text: Accurate

text: dose

text: measurement

text: is

text: particularly

text: important

text: in

text: the

text: dosimetry

text: of

text: small

text: ﬁelds

text: in

text: the

text: presence

text: of

text: low

text: density

text: inhomogeneities.

text: When

text: the

text: photon

text: beams

text: pass

text: through

text: the

text: air

text: cavity,

text: which

text: is

text: the

text: lowest

text: density

text: area

text: in

text: the

text: human

text: body,

text: the

text: dose

text: drops

text: in

text: the

text: air

text: cavity,

text: whereas

text: it

text: gets

text: built-up

text: in

text: the

text: phantom,

text: and

text: thus

text: a

text: dose

text: enhancement

text: may

text: be

text: developed

text: in

text: the

text: distal

text: side

text: of

text: cavity

text: The

text: characteristics

text: of

text: the

text: inhomogeneous

text: conditions

text: were

text: studied

text: with

text: the

text: radiochromic

text: ﬁlm

text: type

text: (Gafchromic

text: type

text: Cat

text: no

text: Lot

text: no

text: by

text: Paelinck

text: et

text: al.

text: They

text: performed

text: the

text: dose

text: measurement

text: with

text: ﬁlms

text: and

text: Monte

text: Carlo

text: simulations

text: for

text: inhomogeneous

text: phantoms

text: involving

text: the

text: air

text: cavity

text: or

text: lung

text: equivalent

text: media

text: with

text: the

text: photon

text: beams

text: provided

text: by

text: linear

text: accelerator.

text: In

text: their

text: simulations,

text: they

text: modeled

text: the

text: phantoms

text: and

text: the

text: ﬁlm

text: by

text: using

text: BEAMnrc/EGSnrc

text: system

text: and

text: then

text: compared

text: the

text: simulation

text: results

text: with

text: measurements,

text: obtaining

text: consistency

text: between

text: the

text: ﬁlm

text: measurements

text: and

text: the

text: calculations

text: with

text: including

text: the

text: ﬁlm.

text: The

text: dose

text: calculations

text: were

text: performed

text: in

text: two

text: diﬀerent

text: methods.

text: First,

text: the

text: simulations

text: were

text: performed

text: with

text: a

text: ﬁlm

text: detector,

text: and

text: thus

text: the

text: doses

text: were

text: obtained

text: from

text: the

text: ﬁlm.

text: We

text: refer

text: to

text: these

text: values

text: of

text: dose

text: as

text: the

text: ”dose

text: to

text: ﬁlm

text: (DTF)”.

text: Second,

text: the

text: simulations

text: were

text: done

text: with

text: phantom

text: geometry

text: without

text: including

text: the

text: ﬁlm.

text: In

text: this

text: case,

text: the

text: absorbed

text: doses

text: were

text: calculated

text: in

text: the

text: phantom

text: material

text: itself

text: without

text: taking

text: into

text: account

text: the

text: presence

text: of

text: the

text: ﬁlm.

text: So,

text: we

text: call

text: these

text: values

text: of

text: dose

text: the

text: ”dose

text: to

text: phantom

text: (DTP)”.

text: For

text: the

text: case

text: when

text: the

text: air

text: cavity

text: was

text: inserted

text: in

text: the

text: phantom,

text: they

text: found

text: a

text: diﬀerence

text: between

text: the

text: DTF

text: and

text: the

text: DTP

text: They

text: showed

text: that

text: the

text: diﬀerences

text: decrease

text: as

text: the

text: ﬁeld

text: size

text: and

text: the

text: density

text: of

text: the

text: medium

text: increase.

text: Also

text: they

text: found

text: that

text: these

text: diﬀerences

text: disappear

text: by

text: oﬀsetting

text: the

text: ﬁlm

text: sideway

text: by

text: a

text: few

text: centimeters.

text: A

text: study

text: of

text: PDD

text: for

text: inhomogeneous

text: phantoms

text: with

text: a

text: Balsa

text: wood

text: as

text: a

text: low

text: density

text: material

text: or

text: a

text: cortical

text: bone

text: as

text: a

text: high

text: density

text: material

text: was

text: done

text: by

text: Wilcox

text: and

text: Daskalov

text: who

text: compared

text: the

text: EBT

text: ﬁlm

text: measurements

text: of

text: PDD

text: (DTF)

text: with

text: the

text: calculations

text: of

text: DTP

text: for

text: inhomogeneous

text: conditions

text: by

text: using

text: Cyberknife

text: systems.

text: In

text: that

text: work,

text: they

text: did

text: not

text: include

text: the

text: presence

text: of

text: the

text: EBT

text: ﬁlm

text: in

text: the

text: simulation

text: since

text: the

text: ﬁlm

text: thickness

text: was

text: smaller

text: than

text: the


text: FIG. 2: (Color online) A schematic diagram of the CyberKnife system used in the simulations. EEDL (Evaluated Electrons Data Library) [? ], EPDL97 (Evaluated Photons Data Library) [? ], and stopping power data [? ? ? ? ]. In our previous study [? ], we modeled a CyberKnife system at the Seoul Uridul Hospital and simulated the photon beams produced by 6 MV CyberKnife using GEANT4 (version 9.2.p01). Figure 2 shows a schematic diagram of the CyberKnife system used in the simu- lations. To make the simulation more eﬃcient, the calculation procedure was split into two steps. The CyberKnife produces electrons impinging on the tungsten target, which creates bremssthlung photons. In the ﬁrst step, we scored the energy and momentum distribution of the photons for 5 to 60 mm cone sizes arriving at the surface of the water phantom of SSD of 80 cm. As the cone sizes vary, diﬀerent energy and momentum distributions are obtained. In the second step, the photon beams were generated at the surface of the water phantom in the direction towards the phantom material by using the energy and angular distributions of the photons as scored in the ﬁrst step. This two step method can save computation time greatly. PDD distributions, output factors and oﬀ axis ratios were also calculated in the second step. In Ref. [? ], we found that our calculated values of these quantities in this two step method agreed with our measured values with errors less than 2%. In this work, we used the same photon beam distribution as scored in Ref. [? ]. We modeled inhomogeneous phantoms with and without the EBT2 ﬁlm. An EBT2 ﬁlm consists 8 ##LTFigure## ##LTFigure##

text: FIG.

text: (Color

text: online)

text: A

text: schematic

text: diagram

text: of

text: the

text: CyberKnife

text: system

text: used

text: in

text: the

text: simulations.

text: EEDL

text: (Evaluated

text: Electrons

text: Data

text: Library)

text: (Evaluated

text: Photons

text: Data

text: Library)

text: and

text: stopping

text: power

text: data

text: In

text: our

text: previous

text: study

text: we

text: modeled

text: a

text: CyberKnife

text: system

text: at

text: the

text: Seoul

text: Uridul

text: Hospital

text: and

text: simulated

text: the

text: photon

text: beams

text: produced

text: by

text: MV

text: CyberKnife

text: using

text: (version

text: Figure

text: shows

text: a

text: schematic

text: diagram

text: of

text: the

text: CyberKnife

text: system

text: used

text: in

text: the

text: simu-

text: lations.

text: To

text: make

text: the

text: simulation

text: more

text: eﬃcient,

text: the

text: calculation

text: procedure

text: was

text: split

text: into

text: two

text: steps.

text: The

text: CyberKnife

text: produces

text: electrons

text: impinging

text: on

text: the

text: tungsten

text: target,

text: which

text: creates

text: bremssthlung

text: photons.

text: In

text: the

text: ﬁrst

text: step,

text: we

text: scored

text: the

text: energy

text: and

text: momentum

text: distribution

text: of

text: the

text: photons

text: for

text: to

text: mm

text: cone

text: sizes

text: arriving

text: at

text: the

text: surface

text: of

text: the

text: water

text: phantom

text: of

text: SSD

text: of

text: cm.

text: As

text: the

text: cone

text: sizes

text: vary,

text: diﬀerent

text: energy

text: and

text: momentum

text: distributions

text: are

text: obtained.

text: In

text: the

text: second

text: step,

text: the

text: photon

text: beams

text: were

text: generated

text: at

text: the

text: surface

text: of

text: the

text: water

text: phantom

text: in

text: the

text: direction

text: towards

text: the

text: phantom

text: material

text: by

text: using

text: the

text: energy

text: and

text: angular

text: distributions

text: of

text: the

text: photons

text: as

text: scored

text: in

text: the

text: ﬁrst

text: step.

text: This

text: two

text: step

text: method

text: can

text: save

text: computation

text: time

text: greatly.

text: PDD

text: distributions,

text: output

text: factors

text: and

text: oﬀ

text: axis

text: ratios

text: were

text: also

text: calculated

text: in

text: the

text: second

text: step.

text: In

text: Ref.

text: we

text: found

text: that

text: our

text: calculated

text: values

text: of

text: these

text: quantities

text: in

text: this

text: two

text: step

text: method

text: agreed

text: with

text: our

text: measured

text: values

text: with

text: errors

text: less

text: than

text: In

text: this

text: work,

text: we

text: used

text: the

text: same

text: photon

text: beam

text: distribution

text: as

text: scored

text: in

text: Ref.

text: We

text: modeled

text: inhomogeneous

text: phantoms

text: with

text: and

text: without

text: the

text: ﬁlm.

text: An

text: ﬁlm

text: consists

text: ##LTFigure##

text: ##LTFigure##


text: 13 [Fe2+, h]0 state (not shown here) is qualitatively and quantitatively similar to that of [Mn2+, h]0. In Fig. ?? we compare the anisotropy energy for the two valence states of Fe, [Fe3+]0 and [Fe2+, h]0, as the impurity atom is moved down from the (110) surface toward the center of the cluster. The qualitative and quantitative diﬀerence between the anisotropy energy in the two valence states is remarkable. While for [Fe2+, h]0 the sublayer dependence shows strong similarities with the case of [Mn2+, h]0, the anisotropy for the the isoelectronic [Fe3+]0 state is quite diﬀerent: it is typically one order of magnitude smaller that for [Fe2+, h]0 and depends weakly on the sublayer. This striking behavior can be easily understood on the basis of the discussion leading to Eqs. ??-??. As for the Mn impurity, the total magnetic anisotropy of the system is closely related to the magnetic anisotropy of the acceptor levels. As seen in Fig. ??, in the isoelectronic [Fe3+]0 state this p-orbital-like acceptor level is occupied by the extra electron that Fe has with respect to Mn. Therefore the total energy of the system contains also the contribution of this level. Then Eq. ?? implies that most of the anisotropy coming from all the other occupied levels of the valance band is essentially canceled by the approximately equal and opposite contribution coming from the occupied acceptor. In contrast, in the [Fe2+, h]0 valence state, the extra electron occupies one of the higher minority d-orbital levels in the gap, and the acceptor level is empty (or is occupied by a ”hole”). Thus, as for the [Mn2+, h]0, the cancellation does not occur. Furthermore, the anisotropy of the top-most d-orbital level is, as expected, small. Therefore the behavior of [Fe2+, h]0 is quite similar to [Mn2+, h]0. These simple considerations lead us to predict that, at least within our − non-self-consistent treatment, the charged states [Fe2+] and [Fe3+, h]+ should have the same behavior of [Fe3+]0 and [Fe2+, h]0 respectively, since the occupancy of the acceptor state is the same. This is exactly what our calculations show. Again, we emphasize that our non-self consistent calculations should be taken cautiously when charged states are involved. The importance of this result stems from the fact that the valence and charge state of individual TM impurities in GaAs can presently be manipulated with a variety of techniques. For example, as mentioned earlier, it is possible to − switch the impurity from the [Fe3+]0 to the [Fe2+] state via a voltage-dependent local manipulation of the Fermi level by means of tip-induced band bending in STM experiments.? A similar manipulation of the Fermi level might soon allow the switching between the [Fe3+]0 and the [Fe3+, h]+ charge state. On the other hand, optical manipulation of the valence state might permit switching between the [Fe3+]0 and the [Fe2+, h]0 states. Our calculations indicate that the switching between these valence and charged states should be also accompanied and characterized by signiﬁcant changes in the magnetic anisotropy energy of the system. We would like to mention that for a Fe impurity in bulk, our ab-initio calculations, in agreement with previously FIG. 14. Color online – Magnetic anisotropy landscape for the Fe dopant in bulk GaAs (a) and Fe on the (110) GaAs surface (b). Energy is in the units of meV. − published results, show that the splitting of the (minority) d levels in the GaAs gap follows the behavior described in Fig. ??. This ﬁgure shows that the dominant eﬀect of the splitting comes from the tetragonal symmetry of the lattice. Eﬀects coming from the spin-orbit interaction, which are certainly present for the d (l = 2) minority-spin level occupied by the sixth electron in Fe, seem to be small. The situation for the Fe dopant on the surface is more complex since, as we have already described, in this case the surface-induced broken symmetry contributes substantially to the splitting, in a way that is not easily disentangled from other, more intrinsic, mechanisms. In any case, it is quite remarkable that the behavior of the anisotropy for the [Fe2+, h]0 complex resembles qualitatively the behavior of the [Mn2+, h]0: in both cases the total anisotropy is controlled by the anisotropy of the impurity acceptor level, regardless of whether or not an extra electron is present. Fig. ?? shows the magnetic anisotropy landscape for the Fe impurity in the isoelectronic state as a function of its position below the surface. Although the results for the surface layer (Fig. ??(b)) and for the ﬁrst sublayer (Fig. ??(a)) are similar, in the latter case the magnetic anisotropy energy is smaller. As the Fe atom is moved away from the surface, the behavior of the magnetic anisotropy approaches that found in the bulk. When Fe is placed in the ninth ##LTFigure## ##LTFigure##

text: state

text: (not

text: shown

text: here)

text: is

text: qualitatively

text: and

text: quantitatively

text: similar

text: to

text: that

text: of

text: In

text: Fig.

text: we

text: compare

text: the

text: anisotropy

text: energy

text: for

text: the

text: two

text: valence

text: states

text: of

text: Fe,

text: and

text: as

text: the

text: impurity

text: atom

text: is

text: moved

text: down

text: from

text: the

text: surface

text: toward

text: the

text: center

text: of

text: the

text: cluster.

text: The

text: qualitative

text: and

text: quantitative

text: diﬀerence

text: between

text: the

text: anisotropy

text: energy

text: in

text: the

text: two

text: valence

text: states

text: is

text: remarkable.

text: While

text: for

text: the

text: sublayer

text: dependence

text: shows

text: strong

text: similarities

text: with

text: the

text: case

text: of

text: the

text: anisotropy

text: for

text: the

text: the

text: isoelectronic

text: state

text: is

text: quite

text: diﬀerent:

text: it

text: is

text: typically

text: one

text: order

text: of

text: magnitude

text: smaller

text: that

text: for

text: and

text: depends

text: weakly

text: on

text: the

text: sublayer.

text: This

text: striking

text: behavior

text: can

text: be

text: easily

text: understood

text: on

text: the

text: basis

text: of

text: the

text: discussion

text: leading

text: to

text: Eqs.

text: As

text: for

text: the

text: Mn

text: impurity,

text: the

text: total

text: magnetic

text: anisotropy

text: of

text: the

text: system

text: is

text: closely

text: related

text: to

text: the

text: magnetic

text: anisotropy

text: of

text: the

text: acceptor

text: levels.

text: As

text: seen

text: in

text: Fig.

text: in

text: the

text: isoelectronic

text: state

text: this

text: p-orbital-like

text: acceptor

text: level

text: is

text: occupied

text: by

text: the

text: extra

text: electron

text: that

text: Fe

text: has

text: with

text: respect

text: to

text: Mn.

text: Therefore

text: the

text: total

text: energy

text: of

text: the

text: system

text: contains

text: also

text: the

text: contribution

text: of

text: this

text: level.

text: Then

text: Eq.

text: implies

text: that

text: most

text: of

text: the

text: anisotropy

text: coming

text: from

text: all

text: the

text: other

text: occupied

text: levels

text: of

text: the

text: valance

text: band

text: is

text: essentially

text: canceled

text: by

text: the

text: approximately

text: equal

text: and

text: opposite

text: contribution

text: coming

text: from

text: the

text: occupied

text: acceptor.

text: In

text: contrast,

text: in

text: the

text: valence

text: state,

text: the

text: extra

text: electron

text: occupies

text: one

text: of

text: the

text: higher

text: minority

text: d-orbital

text: levels

text: in

text: the

text: gap,

text: and

text: the

text: acceptor

text: level

text: is

text: empty

text: (or

text: is

text: occupied

text: by

text: a

text: ”hole”).

text: Thus,

text: as

text: for

text: the

text: the

text: cancellation

text: does

text: not

text: occur.

text: Furthermore,

text: the

text: anisotropy

text: of

text: the

text: top-most

text: d-orbital

text: level

text: is,

text: as

text: expected,

text: small.

text: Therefore

text: the

text: behavior

text: of

text: is

text: quite

text: similar

text: to

text: These

text: simple

text: considerations

text: lead

text: us

text: to

text: predict

text: that,

text: at

text: least

text: within

text: our

text: non-self-consistent

text: treatment,

text: the

text: charged

text: states

text: and

text: h]+

text: should

text: have

text: the

text: same

text: behavior

text: of

text: and

text: respectively,

text: since

text: the

text: occupancy

text: of

text: the

text: acceptor

text: state

text: is

text: the

text: same.

text: This

text: is

text: exactly

text: what

text: our

text: calculations

text: show.

text: Again,

text: we

text: emphasize

text: that

text: our

text: non-self

text: consistent

text: calculations

text: should

text: be

text: taken

text: cautiously

text: when

text: charged

text: states

text: are

text: involved.

text: The

text: importance

text: of

text: this

text: result

text: stems

text: from

text: the

text: fact

text: that

text: the

text: valence

text: and

text: charge

text: state

text: of

text: individual

text: TM

text: impurities

text: in

text: GaAs

text: can

text: presently

text: be

text: manipulated

text: with

text: a

text: variety

text: of

text: techniques.

text: For

text: example,

text: as

text: mentioned

text: earlier,

text: it

text: is

text: possible

text: to

text: switch

text: the

text: impurity

text: from

text: the

text: to

text: the

text: state

text: via

text: a

text: voltage-dependent

text: local

text: manipulation

text: of

text: the

text: Fermi

text: level

text: by

text: means

text: of

text: tip-induced

text: band

text: bending

text: in

text: STM

text: experiments.?

text: A

text: similar

text: manipulation

text: of

text: the

text: Fermi

text: level

text: might

text: soon

text: allow

text: the

text: switching

text: between

text: the

text: and

text: the

text: h]+

text: charge

text: state.

text: On

text: the

text: other

text: hand,

text: optical

text: manipulation

text: of

text: the

text: valence

text: state

text: might

text: permit

text: switching

text: between

text: the

text: and

text: the

text: states.

text: Our

text: calculations

text: indicate

text: that

text: the

text: switching

text: between

text: these

text: valence

text: and

text: charged

text: states

text: should

text: be

text: also

text: accompanied

text: and

text: characterized

text: by

text: signiﬁcant

text: changes

text: in

text: the

text: magnetic

text: anisotropy

text: energy

text: of

text: the

text: system.

text: We

text: would

text: like

text: to

text: mention

text: that

text: for

text: a

text: Fe

text: impurity

text: in

text: bulk,

text: our

text: ab-initio

text: calculations,

text: in

text: agreement

text: with

text: previously

text: FIG.

text: Color

text: online

text: Magnetic

text: anisotropy

text: landscape

text: for

text: the

text: Fe

text: dopant

text: in

text: bulk

text: GaAs

text: (a)

text: and

text: Fe

text: on

text: the

text: GaAs

text: surface

text: (b).

text: Energy

text: is

text: in

text: the

text: units

text: of

text: meV.

text: published

text: results,

text: show

text: that

text: the

text: splitting

text: of

text: the

text: (minority)

text: d

text: levels

text: in

text: the

text: GaAs

text: gap

text: follows

text: the

text: behavior

text: described

text: in

text: Fig.

text: This

text: ﬁgure

text: shows

text: that

text: the

text: dominant

text: eﬀect

text: of

text: the

text: splitting

text: comes

text: from

text: the

text: tetragonal

text: symmetry

text: of

text: the

text: lattice.

text: Eﬀects

text: coming

text: from

text: the

text: spin-orbit

text: interaction,

text: which

text: are

text: certainly

text: present

text: for

text: the

text: d

text: (l

text: minority-spin

text: level

text: occupied

text: by

text: the

text: sixth

text: electron

text: in

text: Fe,

text: seem

text: to

text: be

text: small.

text: The

text: situation

text: for

text: the

text: Fe

text: dopant

text: on

text: the

text: surface

text: is

text: more

text: complex

text: since,

text: as

text: we

text: have

text: already

text: described,

text: in

text: this

text: case

text: the

text: surface-induced

text: broken

text: symmetry

text: contributes

text: substantially

text: to

text: the

text: splitting,

text: in

text: a

text: way

text: that

text: is

text: not

text: easily

text: disentangled

text: from

text: other,

text: more

text: intrinsic,

text: mechanisms.

text: In

text: any

text: case,

text: it

text: is

text: quite

text: remarkable

text: that

text: the

text: behavior

text: of

text: the

text: anisotropy

text: for

text: the

text: complex

text: resembles

text: qualitatively

text: the

text: behavior

text: of

text: the

text: in

text: both

text: cases

text: the

text: total

text: anisotropy

text: is

text: controlled

text: by

text: the

text: anisotropy

text: of

text: the

text: impurity

text: acceptor

text: level,

text: regardless

text: of

text: whether

text: or

text: not

text: an

text: extra

text: electron

text: is

text: present.

text: Fig.

text: shows

text: the

text: magnetic

text: anisotropy

text: landscape

text: for

text: the

text: Fe

text: impurity

text: in

text: the

text: isoelectronic

text: state

text: as

text: a

text: function

text: of

text: its

text: position

text: below

text: the

text: surface.

text: Although

text: the

text: results

text: for

text: the

text: surface

text: layer

text: (Fig.

text: ??(b))

text: and

text: for

text: the

text: ﬁrst

text: sublayer

text: (Fig.

text: ??(a))

text: are

text: similar,

text: in

text: the

text: latter

text: case

text: the

text: magnetic

text: anisotropy

text: energy

text: is

text: smaller.

text: As

text: the

text: Fe

text: atom

text: is

text: moved

text: away

text: from

text: the

text: surface,

text: the

text: behavior

text: of

text: the

text: magnetic

text: anisotropy

text: approaches

text: that

text: found

text: in

text: the

text: bulk.

text: When

text: Fe

text: is

text: placed

text: in

text: the

text: ninth

text: ##LTFigure##

text: ##LTFigure##


text: 4 TABLE I. The on-site energies of Mn and Fe d orbitals in eV. Mn (bulk) Mn (surface) Fe (bulk) Fe (surface) eup -4.5 -4.5 -6.5 -4.5 g tup -2.226 -2.226 -6.5 -4.5 2g edn 3.5 3.5 -0.138 -1 g tdn 3.5 3.5 -0.02 -1 2g In Fig. 3 we also plot the spin-resolved density of states for the d-orbitals of a Ga atom on the (110) GaAs surface, 4 U=0 dx2−y2+dz2 2 d +d +d xy yz zx 0 −2 V OS/e−145 D dx2−y2+dz2 U=4 eV 10 d +d +d xy yz zx 5 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 Energy (eV) FIG. 2. Color online – Spin-resolved DOS for the d-orbitals of the Mn dopant on the (110) GaAs surface for U = 0 (top panel) and U = 4 eV (bottom panel). The vertical dashed line at E = 0 denotes the position of the Fermi energy. and compare it with the analogous quantity for a substitutional Mn atom. We can clearly see that the Ga d-orbitals are, as expected, unpolarized and occur at energies far below the Fermi level. The band structure of bulk GaAs (see U=0 10 U=4 eV V e S/ 0 O D −10 Ga d Mn d Ga d Mn d up up dn dn −15 −10 −5 0 5 Energy (eV) FIG. 3. Color online – Spin-resolved density of states (DOS) for the d-orbitals of a Ga atom and that of a substitutional Mn on the (110) GaAs surface. The vertical dashed line at E = 0 denotes the position of the Fermi level. − Fig. 4) reveals that the top of the valence band in GaAs is completely dominated by p like states, both for Ga and − As. Levels of d character are visibly present, particularly for Ga atoms, even at the Gamma point. However, their − contribution is only 2% of the p like states. These results justify our simplifying choice of disregarding Ga d-orbitals − in the TB model. We would like to emphasize that the d level parameters in the empirical TB model can diﬀer considerably from the corresponding electronic atomic orbitals. Therefore our remark on the position of the ab-initio − Ga d level energy being far away from the Fermi energy as a justiﬁcation of our neglect of these levels in the TB model must be taken with caution. Note, however, that the parametrization of GaAs that we adopt here, which − excludes d levels of Ga and As atoms, is a standard procedure and therefore the ab-initio results are not inconsistent ##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##

text: TABLE

text: I.

text: The

text: on-site

text: energies

text: of

text: Mn

text: and

text: Fe

text: d

text: orbitals

text: in

text: eV.

text: Mn

label: table

text: (bulk)

text: Mn

text: (surface)

text: Fe

text: (bulk)

text: Fe

text: (surface)

text: eup

text: g

text: tup

text: edn

text: g

text: tdn

text: In

text: Fig.

text: we

text: also

text: plot

text: the

text: spin-resolved

text: density

text: of

text: states

text: for

text: the

text: d-orbitals

text: of

text: a

text: Ga

text: atom

text: on

text: the

text: GaAs

text: surface,

font: default

text: d

text: +d

text: +d

text: xy

text: yz

text: zx

text: −2

text: V

text: D

text: eV

text: d

text: +d

text: +d

text: xy

text: yz

text: zx

text: −5

text: −4

text: −3

text: −2

text: −1

text: Energy

text: (eV)

text: FIG.

text: Color

text: online

text: Spin-resolved

text: DOS

text: for

text: the

text: d-orbitals

text: of

text: the

text: Mn

text: dopant

text: on

text: the

text: GaAs

text: surface

text: for

text: U

text: (top

text: panel)

text: and

text: U

text: eV

text: (bottom

text: panel).

text: The

text: vertical

text: dashed

text: line

text: at

text: E

text: denotes

text: the

text: position

text: of

text: the

text: Fermi

text: energy.

text: and

text: compare

text: it

text: with

text: the

text: analogous

text: quantity

text: for

text: a

text: substitutional

text: Mn

text: atom.

text: We

text: can

text: clearly

text: see

text: that

text: the

text: Ga

text: d-orbitals

text: are,

text: as

text: expected,

text: unpolarized

text: and

text: occur

text: at

text: energies

text: far

text: below

text: the

text: Fermi

text: level.

text: The

text: band

text: structure

text: of

text: bulk

text: GaAs

text: (see

text: eV

text: V

text: e

text: S/

text: O

text: D

text: Ga

text: d

text: Mn

text: d

text: Ga

text: d

text: Mn

text: d

text: up

text: up

text: dn

text: dn

text: −5

text: Energy

text: (eV)

text: FIG.

text: Color

text: online

text: Spin-resolved

text: density

text: of

text: states

text: (DOS)

text: for

text: the

text: d-orbitals

text: of

text: a

text: Ga

text: atom

text: and

text: that

text: of

text: a

text: substitutional

text: Mn

text: on

text: the

text: GaAs

text: surface.

text: The

text: vertical

text: dashed

text: line

text: at

text: E

text: denotes

text: the

text: position

text: of

text: the

text: Fermi

text: level.

text: Fig.

text: reveals

text: that

text: the

text: top

text: of

text: the

text: valence

text: band

text: in

text: GaAs

text: is

text: completely

text: dominated

text: by

text: p

text: like

text: states,

text: both

text: for

text: Ga

text: and

text: As.

text: Levels

text: of

text: d

text: character

text: are

text: visibly

text: present,

text: particularly

text: for

text: Ga

text: atoms,

text: even

text: at

text: the

text: Gamma

text: point.

text: However,

text: their

text: contribution

text: is

text: only

text: of

text: the

text: p

text: like

text: states.

text: These

text: results

text: justify

text: our

text: simplifying

text: choice

text: of

text: disregarding

text: Ga

text: d-orbitals

text: in

text: the

text: TB

text: model.

text: We

text: would

text: like

text: to

text: emphasize

text: that

text: the

text: d

text: level

text: parameters

text: in

text: the

text: empirical

text: TB

text: model

text: can

text: diﬀer

text: considerably

text: from

text: the

text: corresponding

text: electronic

text: atomic

text: orbitals.

text: Therefore

text: our

text: remark

text: on

text: the

text: position

text: of

text: the

text: ab-initio

text: Ga

text: d

text: level

text: energy

text: being

text: far

text: away

text: from

text: the

text: Fermi

text: energy

text: as

text: a

text: justiﬁcation

text: of

text: our

text: neglect

text: of

text: these

text: levels

text: in

text: the

text: TB

text: model

text: must

text: be

text: taken

text: with

text: caution.

text: Note,

text: however,

text: that

text: the

text: parametrization

text: of

text: GaAs

text: that

text: we

text: adopt

text: here,

text: which

text: excludes

text: d

text: levels

text: of

text: Ga

text: and

text: As

text: atoms,

text: is

text: a

text: standard

text: procedure

text: and

text: therefore

text: the

text: ab-initio

text: results

text: are

text: not

text: inconsistent

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: 6 5 e g t 2g 0 U=0 eV−5 S/ 5 O e D U=4 eV g t 2g 0 −5 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 Energy(eV) FIG. 5. Color online – Spin-resolved DOS for the d-orbitals of the Fe dopant in bulk GaAs for U = 0 (top panel) and U = 4 eV (bottom panel). The vertical dashed line at E = 0 denotes the position of the Fermi level. 2 0 −2 dx2−y2+dz2 U=0 V−4 dxy+dyz+dzx e S/ 4 O D U=4 eV dx2−y2+dz2 2 d +d +d xy yz zx 0 −2 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 4 Energy (eV) FIG. 6. Color online – Spin-resolved DOS for the d-orbitals of the Fe dopant on the (110) GaAs surface for U = 0 (top panel) and U = 4 eV (bottom panel). The vertical dashed line at E = 0 denotes the position of the Fermi level. A. Mn dopants on (110) GaAs surface We start by discussing the results of TB modeling of a single Mn dopant in GaAs. As we will see in the following, our TB model incorporating the impurity d-orbitals reproduces the well-known features of Mn in the bulk and on the surface of GaAs, in agreement with previous studies. However, we also ﬁnd some interesting diﬀerences between our approach and the classical-spin model.? These diﬀerences are mainly related to the magnetic anisotropy of Mn on the surface and in the subsurface layers. Fig. 7 shows the electronic properties of Mn in the bulk (top panels) and on the (110) surface (bottom panels) of GaAs. Mn introduces three levels in the GaAs gap, with the highest level, which is unoccupied, known as the hole- acceptor level. The other two levels are occupied and they lie below the acceptor. The position of the acceptor level with respect to the valence band is found at 113 meV for the bulk and at 0.89 eV for the surface dopant. While the bulk calculation reproduces exactly the experimental value? ? ? ? (see the discussion about the TB parametrization in section ), the surface calculation also gives the position of the acceptor level close to the experimental result.? As one can see from Fig. 7(c), the calculated LDOS for the acceptor on the surface shows more concentration of the spectral weight on the Mn site compared to the bulk case, which is an indication of a deeper and a more localized state. In general, the calculations presented in Fig. 7 support the results of the classical spin model? ? and are in good agreement with other theoretical and experimental results.? ? We would like to comment on one important feature of the electronic structure of Mn in bulk GaAs. According to Fig. 7(a) the three levels introduced by Mn in the bulk GaAs gap are found to be spread over an energy interval of ##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## ##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##

text: e

text: g

text: t

text: S/

text: O

text: e

text: D

text: eV

text: g

text: t

text: −5

text: −7

text: −6

text: −5

text: −4

text: −3

text: −2

text: −1

text: Energy(eV)

text: FIG.

text: Color

text: online

text: Spin-resolved

text: DOS

text: for

text: the

text: d-orbitals

text: of

text: the

text: Fe

text: dopant

text: in

text: bulk

text: GaAs

text: for

text: U

text: (top

text: panel)

text: and

text: U

text: eV

text: (bottom

text: panel).

text: The

text: vertical

text: dashed

text: line

text: at

text: E

text: denotes

text: the

text: position

text: of

text: the

text: Fermi

text: level.

text: −2

text: dxy+dyz+dzx

text: e

text: S/

text: O

text: D

text: eV

text: d

text: +d

text: +d

text: xy

text: yz

text: zx

text: −2

text: −7

text: −6

text: −5

text: −4

text: −3

text: −2

text: −1

text: Energy

text: (eV)

text: FIG.

text: Color

text: online

text: Spin-resolved

text: DOS

text: for

text: the

text: d-orbitals

text: of

text: the

text: Fe

text: dopant

text: on

text: the

text: GaAs

text: surface

text: for

text: U

text: (top

text: panel)

text: and

text: U

text: eV

text: (bottom

text: panel).

text: The

text: vertical

text: dashed

text: line

text: at

text: E

text: denotes

text: the

text: position

text: of

text: the

text: Fermi

text: level.

text: A.

text: Mn

text: dopants

text: on

text: GaAs

text: surface

text: We

text: start

text: by

text: discussing

text: the

text: results

text: of

text: TB

text: modeling

text: of

text: a

text: single

text: Mn

text: dopant

text: in

text: GaAs.

text: As

text: we

text: will

text: see

text: in

text: the

text: following,

text: our

text: TB

text: model

text: incorporating

text: the

text: impurity

text: d-orbitals

text: reproduces

text: the

text: well-known

text: features

text: of

text: Mn

text: in

text: the

text: bulk

text: and

text: on

text: the

text: surface

text: of

text: GaAs,

text: in

text: agreement

text: with

text: previous

text: studies.

text: However,

text: we

text: also

text: ﬁnd

text: some

text: interesting

text: diﬀerences

text: between

text: our

text: approach

text: and

text: the

text: classical-spin

text: model.?

text: These

text: diﬀerences

text: are

text: mainly

text: related

text: to

text: the

text: magnetic

text: anisotropy

text: of

text: Mn

text: on

text: the

text: surface

text: and

text: in

text: the

text: subsurface

text: layers.

text: Fig.

text: shows

text: the

text: electronic

text: properties

text: of

text: Mn

text: in

text: the

text: bulk

text: (top

text: panels)

text: and

text: on

text: the

text: surface

text: (bottom

text: panels)

text: of

text: GaAs.

text: Mn

text: introduces

text: three

text: levels

text: in

text: the

text: GaAs

text: gap,

text: with

text: the

text: highest

text: level,

text: which

text: is

text: unoccupied,

text: known

text: as

text: the

text: hole-

text: acceptor

text: level.

text: The

text: other

text: two

text: levels

text: are

text: occupied

text: and

text: they

text: lie

text: below

text: the

text: acceptor.

text: The

text: position

text: of

text: the

text: acceptor

text: level

text: with

text: respect

text: to

text: the

text: valence

text: band

text: is

text: found

text: at

text: meV

text: for

text: the

text: bulk

text: and

text: at

text: eV

text: for

text: the

text: surface

text: dopant.

text: While

text: the

text: bulk

text: calculation

text: reproduces

text: exactly

text: the

text: experimental

text: value?

text: (see

text: the

text: discussion

text: about

text: the

text: TB

text: parametrization

text: in

text: section

text: the

text: surface

text: calculation

text: also

text: gives

text: the

text: position

text: of

text: the

text: acceptor

text: level

text: close

text: to

text: the

text: experimental

text: result.?

text: As

text: one

text: can

text: see

text: from

text: Fig.

text: the

text: calculated

text: LDOS

text: for

text: the

text: acceptor

text: on

text: the

text: surface

text: shows

text: more

text: concentration

text: of

text: the

text: spectral

text: weight

text: on

text: the

text: Mn

text: site

text: compared

text: to

text: the

text: bulk

text: case,

text: which

text: is

text: an

text: indication

text: of

text: a

text: deeper

text: and

text: a

text: more

text: localized

text: state.

text: In

text: general,

text: the

text: calculations

text: presented

text: in

text: Fig.

text: support

text: the

text: results

text: of

text: the

text: classical

text: spin

text: model?

text: and

text: are

text: in

text: good

text: agreement

text: with

text: other

text: theoretical

text: and

text: experimental

text: results.?

text: We

text: would

text: like

text: to

text: comment

text: on

text: one

text: important

text: feature

text: of

text: the

text: electronic

text: structure

text: of

text: Mn

text: in

text: bulk

text: GaAs.

text: According

text: to

text: Fig.

text: the

text: three

text: levels

text: introduced

text: by

text: Mn

text: in

text: the

text: bulk

text: GaAs

text: gap

text: are

text: found

text: to

text: be

text: spread

text: over

text: an

text: energy

text: interval

text: of

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: 9 FIG. 9. Color online – The (110) surface LDOS as a function of Mn depth. Panels (a) to (i) correspond to the LDOS calculated for Mn in sublayers one to nine, respectively. The left and the right columns show the LDOS for easy and hard direction, respectively. Mn is successively moved down from the surface layer towards the center of the cluster. Fig. ?? shows a comparison of the anisotropy energy calculated with the two models as a function of the Mn depth, where we also include the values for the (110) surface and for the bulk. Here the value of the anisotropy energy is deﬁned to be diﬀerence between the maximum and the minimum ground-state energy, calculated as a function of either the direction of magnetic moment in the case of classical-spin model, or the quantization axis in the quantum d-level model. We would like to point several important features that can be seen on this ﬁgure. The lower value of the anisotropy energy compared to the classical-spin model, found in our calculations, persists for almost all of the considered subsurfaces. The relatively high anisotropy for the bulk found in both models is a ﬁnite-size eﬀect. In fact, our calculations demonstrate that the bulk anisotropy energy drops to a small fraction of a meV when the number of atoms in the cluster is increased ##LTFigure## ##LTFigure##

text: FIG.

text: Color

text: online

text: The

text: surface

text: LDOS

text: as

text: a

text: function

text: of

text: Mn

text: depth.

text: Panels

text: (a)

text: to

text: (i)

text: correspond

text: to

text: the

text: LDOS

text: calculated

text: for

text: Mn

text: in

text: sublayers

text: one

text: to

text: nine,

text: respectively.

text: The

text: left

text: and

text: the

text: right

text: columns

text: show

text: the

text: LDOS

text: for

text: easy

text: and

text: hard

text: direction,

text: respectively.

text: Mn

text: is

text: successively

text: moved

text: down

text: from

text: the

text: surface

text: layer

text: towards

text: the

text: center

text: of

text: the

text: cluster.

text: Fig.

text: shows

text: a

text: comparison

text: of

text: the

text: anisotropy

text: energy

text: calculated

text: with

text: the

text: two

text: models

text: as

text: a

text: function

text: of

text: the

text: Mn

text: depth,

text: where

text: we

text: also

text: include

text: the

text: values

text: for

text: the

text: surface

text: and

text: for

text: the

text: bulk.

text: Here

text: the

text: value

text: of

text: the

text: anisotropy

text: energy

text: is

text: deﬁned

text: to

text: be

text: diﬀerence

text: between

text: the

text: maximum

text: and

text: the

text: minimum

text: ground-state

text: energy,

text: calculated

text: as

text: a

text: function

text: of

text: either

text: the

text: direction

text: of

text: magnetic

text: moment

text: in

text: the

text: case

text: of

text: classical-spin

text: model,

text: or

text: the

text: quantization

text: axis

text: in

text: the

text: quantum

text: d-level

text: model.

text: We

text: would

text: like

text: to

text: point

text: several

text: important

text: features

text: that

text: can

text: be

text: seen

text: on

text: this

text: ﬁgure.

text: The

text: lower

text: value

text: of

text: the

text: anisotropy

text: energy

text: compared

text: to

text: the

text: classical-spin

text: model,

text: found

text: in

text: our

text: calculations,

text: persists

text: for

text: almost

text: all

text: of

text: the

text: considered

text: subsurfaces.

text: The

text: relatively

text: high

text: anisotropy

text: for

text: the

text: bulk

text: found

text: in

text: both

text: models

text: is

text: a

text: ﬁnite-size

text: eﬀect.

text: In

text: fact,

text: our

text: calculations

text: demonstrate

text: that

text: the

text: bulk

text: anisotropy

text: energy

text: drops

text: to

text: a

text: small

text: fraction

text: of

text: a

text: meV

text: when

text: the

text: number

text: of

text: atoms

text: in

text: the

text: cluster

text: is

text: increased

text: ##LTFigure##

text: ##LTFigure##