Magnetization of type-II superconductors in the modified Bean model

PHYSICA ELSEVIER

Physica C 268 (1996) 241-256

Magnetization of type-II superconductors in the modified Bean model Shinjiro Tochihara *, Hiroshi Yasuoka, Hiromasa Mazaki Departmentof Mathematics and Physics, The NationalDefenseAcademy, Yokosuka239, Japan Received 18 June 1996; revised manuscript received 4 July 1996

Abstract Within the construct of the modified Bean model which takes into consideration the surface barrier A H and a nonzero value of the lower critical field Hcl, we have calculated the initial magnetization curves and full hysteresis loops of type-II superconductors immersed in an external field H = Hdc + Ha¢ cos(tot), where Hat ( > 0) is a d c bias field and H~ ( > 0) is an ac field amplitude. We denote the maximum and minimum values of H by HA(= Hdc + Ha¢) and Ha( = Hd~ -- Hat). We consider an infinitely long cylinder with radius a, and the applied field along the cylinder axis. Magnetization equations M(H) for full hysteresis loops are derived for four different ranges of HA:

O
HcI+AH
HcI+AH+Hp
HcI-FAH+2Hp
Here Hp is the field for full penetration. Each of these four cases is further classified for several ranges of H B. To describe completely the descending and ascending branches of full hysteresis loops for all cases, 83 stages of H are considered. To verify the present derivations, all the equations were confirmed to be continuous at their end points. Some typical hysteresis loops computed using the appropriate magnetization equations are demonstrated. From the results, we recognize the role of A H and H ci on the hysteresis loops. A H does not cause any essential deformation of the M ( H ) curves, but merely expands them up and down with the increase of AH. On the contrary, Hcl introduces a step-like feature into the hysteresis loops, resulting in drastic change in their shape. Present derivations would be a useful tool for analyses of the magnetization curves of type-II superconductors.

I. Introduction The critical-state model first introduced by Bean [1,2] and London [3] has been widely used for study of magnetic properties o f type-II superconductors. The essential aspect o f this model is that, when a

* Corresponding author. Fax: +81 468 44 5902.

magnetic field is applied to a sample, a macroscopic supercurrent circulates in the sample with a criticalcurrent density Jc (Bi), where B i is the local flux density inside the specimen. The simplest case proposed by Bean [1] is that Jc is a constant independent o f B i. Despite o f the simplified treatment, B e a n ' s model has been proved to be very useful for investigations of the magnetic

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S. Tochihara et a l . / Physica C 268 (1996) 241-256

242

response of superconductors. On the assumption that the critical-current density has the form

Yc = k/(Bo + I

I),

Hcj on the magnetic behavior of type-II superconductors can be plainly seen.

(1)

where k and B 0 are constants, Kim, Hempstead and Stmad [4,5] and Anderson [6] studied the critical phenomena of type-II superconductors. As pointed out by Chen and Goldfarb [7], the relation given by Eq. (1) is a generalized form of the critical-state model, because it leads to either the linear model [8] or to the Bean model [2] or to the power-law model [9,10], depending on the relative magnitude of k, B0 and B i . In a series of investigations of magnetic properties of ceramics superconductors immersed in an ac magnetic field superimposed on a dc field, we extensively applied the Kim-Anderson model and found that the magnetization equations based on this model qualitatively reproduce well the observations [ 11 - 16]. However, in these previous treatments, we have neither taken into consideration the surface barrier A H nor the lower critical field Hc~, i.e., both of these were assigned to be zero. Even for the case A H = Hcl = 0, it is required to derive the magnetization equations of 58 stages to bring full hysteresis loops to completion [17]. The role of AH a n d / o r Hc~ in the magnetization of type-II superconductors has been studied by many workers [18-29]. However, to the authors' knowledge, complete derivations of the magnetization equations including A H and Hcl have not so far been published. Motivated by their work, within the construct of the modified Bean model, we attempted to derive fully the magnetization equations of type-II superconductors immersed in an alternating field superimposed on a dc field, H ( t o t ) = H d c + H a c cos(tot), where Hd~ (_> 0) is a dc basis field and Hac ( > 0) is an ac field amplitude. Both A n and Hcl certainly depend on the local flux density. However, in the present work, we assumed both of them are constants, similar to JcIn the following, we describe the details of the derivation. To complete full hysteresis loops, one needs to consider 83 stages of n . Using these equations, we calculate M(H) curves and verify that the curves are continuous at their end points. Through the calculated M(H) curves, the effect of A H and

2. General expressions for magnetization for an infinite cylinder According to Dunn and Hlawiczka [20], the magnetization curve is somewhat complicated due to the nonzero values of A H and Hc~, but the effective field in the sample Heff may be described by the expression [20] nef f =

H n - -~[ncl

dH/dt IdH/dt-------~l An,

(2)

where H is the applied field. Adopting their proposal, we derive the general expression for magnetization. We consider an infinite cylindrical specimen with radius a, where the boundary of the sample is at x = a. An external field H is applied along the axis. In this configuration, the local magnetic flux density is expressed as Bi(x). In the critical-state model, by applying an external field H, macroscopic supercurrent J ( x ) flows in the sample and B~(x) is written as

J;J(x')

Bi(x ) =/x0Heff+/.t 0

dx',

(3)

where ~o = 47r× 10 - 7 H / m . Using the critical current Jc (positive constant) and Ampere's law, V× B = / x 0 J , we obtain

dBi( x ) / d x = -/x0J (x) = -sgn(J)

/XoJ~,

(4)

where the sign function sgn(X) is 1 if X > 0, - 1 if X < 0, and 0 if X = 0. After integration, we obtain Bi( x)/ix o = - s g n ( J ) Jc x + c,

(5)

where c is an integration constant to be determined by the boundary conditions. From the boundary condition Bi(a)/~ o = Heff, the constant c is given as C ~---n e f f -t-

s g n ( J ) Jc a.

(6)

Substituting Eq. (6) into Eq. (5), we obtain

Bi( x)/ o = - sgn( J ) Jc x + Heff + s g n ( J ) Jc a.

(7)

S. Tochihara et aL / Physica C 268 (1996) 241-256

243

Half

Using B~(x), we obtain the average flux density B ( H ) in the sample,

8(n) /x0

2 fa 8,(x)

=

a--5- o x

/%

dx.

(a)

(8) et-

Thus, the average magnetization M ( H ) of the sample is given by

X

0

(9)

M ( H ) = B( H ) / ~ o - H.

Half

3. Initial magnetization and full-penetration field

:~ ~

2Hp

3.1. Full-penetration field Hp

We start from the initial state, H = Bi(x) = 0, and increase H in the direction of the cylinder axis. According to Lenz's law, when the applied field exceeds H = H~t + A H, the supercurrent J (of negative sign) begins to penetrate from the sample surface ( x = a) inward. If the supercurrent penetrates until x = x 0, J ( x ) is given as J(x) =0

(10a)

(O
J ( x ) = - J c ( x0 < x < a).

0

-"~--X~

a

Fig. 1. Schematic representations of the full-penetration field Hp. x o in the ascending branch and x t in the descending branch are given by Eqs. 0 2 ) and (29), respectively, a is the cylinder surface.

(10b)

In the region specified by Eq. (10a), B i ( x ) / / p , because there exists no supercurrent. Using Eqs. (7) and (10b), Bi(x) is given as

0 =

0 For the first case (0 < H < Hot + AH), the distribution of flux density in the sample is

Bi( x ) / / z 0 = J c x + n e f f - J c a (Xo
(ll)

From the boundary condition Bi( x o ) / i z o = O, x o can be obtained from Eq. (1 l) as x o = a - neff/J c.

(12)

Since the full-penetration field np is Herf for x 0 = 0 [see Fig. la], we obtain H p = J c a.

(13)

3.2. Local flux density and B(H)

We consider the magnetization for three cases: O
Bi(x)/txo=O

(14)

(O
leading to the trivial result, B ( H ) / I x o = O. For the second case (Hcl + A H < H < He1 + AH + n . ) , neff is given a s n e f f - - H - n c l - A n f r o m Eq. (2). The local flux density in the region 0 < x < x 0 is of course zero. From Eqs. (11) and (13), the local flux density is given as

Bi(x)

bo(x)

/x0

/x0 = (npla)x+n-nc,

- An-rip

(Xo x a),

(15)

Hcl + A H < H
where bo(x) is defined by Bi(x) derived using the flowing supercurrent in the region x o < x < a.

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S. Tochihara et a l . / Physica C 268 (1996) 241-256

Using Eq. (8), the average flux density for this case is B(H)

-IXo

2

=--ix a2

.a b 0 ( x )

x

o

tZo

--~"0.

dx

(H-Hcl-AH):

(H-Hc,-AH)

Hp

3

3 Hp2 (16)

For the third case (Hc~ + A H + Hp < H), the distribution of flux density in the sample is Bi(x)

/x0

Hp x + n a

-

ncl

-

For the first case (0 < H A < Hcl + A H ) , full hysteresis loops have two stages, ascending and descending. In both the stages, B i ( x ) = 0, i.e., B ( H ) The average flux density in the other three cases are discussed below. Note that when a dc bias field Hd¢ has a nonzero value, each of these cases is further classified into several different cases, depending on the magnitude of H B.

4.1. Hysteresis loops for the low-Ha case (Her + A H < HA < Hc, + A H + H p) In the initial magnetization process, B ( H ) for H = H A is obtained from Eq. (16):

AH -np (17)

B(B)

Using Eq. (8), the average flux density for this case is

iZo

(0 < x < a ) .

B( n ) / / , t 0 - a - Be, - A n - 3np.

(H^-Hel-AH)

2

Hp

-

(18)

(HA-He,--AH) 3Hp2

3 (19)

At this field, using Eqs. (12) and (13), we find

H p - H A +HcI + A H

a,

(20)

4. Full hysteresis loops

X0A ~---

To obtain full hysteresis loops, we consider a period of the applied field H = Hdc + Hae COS(tOt), where //de > 0 and Ha¢ > 0. Denoting the maximum and minimum values of H by HA(= Hd¢ +Hac) and Ha( =//de - H a c ) , respectively, four types of hysteresis loops appear, depending on the magnitude of H A. The first case is for 0 < H A ~ H c i + A H, where no magnetic field penetrates into the specimen. The second case is for H c I + A H < H A ~ H e I + A H + Hp, where the specimen is never fully penetrated. The third case is for Hcl + A H + 2Hp < H A, when the reverse supercurrent penetrates to the center of the specimen before Herf is cycled back to zero (see Fig. lb). The fourth case is intermediate, Hcl + A H + H p < H A < H c t + A H + 2 H p. When a dc bias field Ha~ has a nonzero value, each of these cases is further classified into several different cases, depending on the magnitude of H B. Full hysteresis loops are derived by decreasing H from H A to H B, forming the descending branch of the loops, and then by increasing H from H B to H A, forming the ascending branch.

where an additional subscript A indicates the specified variable or function at H = H A. For simplicity, in the following discussion, we use nef f (n) (n = 1, 2, 3, 4) defined below:

np

neff(1 ) = n - n e l -- A n ,

(21a)

Heff(2 ) = H - Hcl + A H ,

(21b)

neff(a ) = n + ncl + An,

(21c)

neff(n )

(21d)

=

H + Her - A n .

4.1.1. For H B < -He1 - A H According to Dunn and Hlawiczka [20], at the turning points of the applied field H ( [ H I > Hel) surface sheath currents screen the specimen and cause flux to be completely trapped as H decreases (or increases) over a range 2AH. This results inevitably in complete flux trapping for applied fields in the range - H e I - T - A H < H < H eI-T-AH (minus: descending; plus: ascending). Stage 1. HA - 2 A H < H < H g (descending). In this stage, the flux is completely trapped. The effec-

S. Tochiharaet aL/ PhysicaC 268 (1996)241-256 five field Heff is Heff (1A). The supercurrent J (=-J¢) penetrates from the sample surface until x = X0A. THUS, we obtain the local flux density b0A(X) from Eq. (15) as boa (X)

Hp

= - - X + H A - Hel - A H - Hp a

/x 0

( x0A < x < a ) .

B( H) =--2 f a boA(X ) dx I.LO a 2 ..XoA tZo Hp

3Hp2

q0 = Heft( 1A ), qi = neff( 1A ) -- Heff(2), q2 = Heff( 1A ) -- Heff(3), q3 = Heff(3a) - Heff(4), q4 = H~ff(3a) -- Heel(1), q5 =/-/eff(2B) -- neff(1), q6 = Heff(3) -- Heff(1),

G,(qm) = q2/2Hp, q3/12n~,

63(qm) =

q~/Hp,

G4(qm) =

q3 / 3 H2p•

Hp

/x o

a

(23)

(24a) (24b) (24c) (240) (24e) (24f) (24g) (25a) (25b) (25c) (250)

=

B( H)/tXo = G3(qo) - (74( qo)(27) Stage 2. Hcl -- A H < H < H A - 2 A H (descending). In this stage, the supercurrent J ( = Jc) penetrates from the sample surface until x = x~ and the

(29)

: X ,o - - G l ( q l ) + a 2 ( q l ) + G3(qo)-

G4(qo).

Stage 3. - H e ¿ - A H < H < Hcl - A H (descending). In this stage, the flux is completely trapped. The supercurrent J ( = J~) penetrates from the sample surface until x = xl z and the effective field Heff is zero. From Eq. (7), the local flux density blz(X) is expressed by blz ( x) /iZo = - ( Hp/a)x + Hp (Xlz
(31)

where an additional subscript z on b l ( X ) and x 1 indicates the specified values at Heyf = 0. For example,

qlz = Heff(1A)

----HA -- Hcl -- A H .

Using the boundary condition at = boA(Xi z), we obtain

(32)

x = x l z , blz(xlz)

- n A + Hcl + A n + 2np

B(H)

Using Eq. (25), Eq. (23) can be rewritten as

x = x I, b0(xl) =

(30)

q2B = Heff( 1g ) -- Heff(aB) (26)

(28)

B(H) is

Thus,

HA -- HB -- 2Hct - 2 A H .

+AH+Hp

H - H A + 2AH + 2H p a. 2up

Similar to the subscript A, an additional subscript B indicates the specified variable or function at H = H a . For example,

=

x+H-Hel

Using the boundary condition at bl(xl), we obtain

Thus,

For further simplicity, in the following discussion, we use qm (m = 0, l, 2 . . . . . 6) in a specified region of x for each m and we define Gt(q,,,) (l = 1, 2, 3, 4) as below:

G2(q,,) =

b,( x)

X 1 ~--

(noff(1A)) 3

-

effective field n e f f is Heff (2). From Eq. (7), bl(x), which is assigned to Bl(X) in the region x t < x < a, is expressed by

( x, < x < a ) . (22)

Using E q . ( 1 6 ) , we obtain

(neff(1A)) 2

245

X~z =

a.

2Hp

(33)

B(H) is

tZo 2 [ .x.

a---'ik JXo^

bOA( x )

tXo

a

d X + fx X

blz( X)xto dx )

= - G , ( q , z ) + G2(q,z ) + G3(qo) - G4(qo ). (34)

246

S. Tochihara et al./ Physica C 268 (1996) 241-256

Stage 4. H a ~ H < - H c i -- A H (descending). In this stage, the supercurrent J ( = Jc) penetrates from the sample surface until x = x 2 and the effective field Herf is Hal (3). From Eq, (7), the local flux density b2(x) is expressed by

Stage 6. H a + 2 A H < H < --Hcl + A H (ascending). In this stage, the supercurrent J ( = - J c ) penetrates from the sample surface until x = x 3 and the effective field Heft is Heft (4). From Eq. (7), the local flux density b3(x) is expressed by

b2(x.....~) = -__HP x + H + Hcl + AH + Hp /-to a

b3(X)

( x 2 < x < a).

Hp

--x+H+Hcl-AH-Hp a

=

tXo

(35) (x3
(41)

Using the boundary condition at x = x 2, b2(x 2) = b0A(X2), we obtain H - H A + 2Hcl + 2 A H + 2 H p a. (36) x2 = 2 Hp

Using the boundary condition at x = x 3, b3(x3)= b2a(X3), we obtain

Thus, B ( H ) is

X3 =

B( H ) = ~2

/-t'0

dx+

x2x

a2 / XoA

dx

H a - H + 2AH + 2Hp a. 2Hp

Thus, B ( H ) is

~0

= - G , ( q 2 ) + G2(q2 ) + G3(qo) - G4(qo ). (37)

B( H )

2

hi'0

dx

x2B

~o

= --Gl(q2B) + G2(q2B ) + G,(q3)

a

+ G2(q3) + G3(qo) - G4( qo)"

+ HB +H¢ + A H + H p

(38) Using the boundary condition at x = X2B, b2a(X2B) = b0A(X2B), we obtain H B - H A +2He, + A H + 2 H p a. (39) x2a = 2 Hp

Hp =

tZo

B(H)

~

X

--

Hp

a

( x3z < x < a ) ,

/Zo

(x) dx)

2 [ [X2BxbOA( X ) dx'Jt" / a xb2B aL ~JXoA l~O Xia ~'£0

(44)

Using the boundary condition at x = x3z, b3z(x3z) = b2B(Z3z) , we obtain H B + Hcl

--GI(q2B) + G2(q2B) + G3(qo) - G4(qo ).

(40)

(43)

Stage 7. - H c i + A H < H < H¢ l + A H (ascending). In this stage, the flux is completely trapped. The supercurrent J ( = - J c ) penetrates from the sample surface until x = x3z and the effective field Herf is zero. From Eq. (7), the local flux density b3z(X) is expressed by b3z ( x )

Thus, B ( H ) is

=

/'tO

+ / X 3 x b 2 a ( X) d x

np

/z o

f x2, x bOA(X)

a2 k XOA

Stage 5. H a < H < H a + 2 A H (ascending). In this stage, the flux is completely trapped. The supercurrent J ( = Jc) penetrates from the sample surface until x = x2a and the effective field Herf is Heef (3s). From Eq. (7), the local flux density b2B (x) is expressed by

b2B( x )

(42)

X3z =

+ A H + 2Hp 2Hp

a.

(45)

S. Tochihara et al. / Physica C 268 (1996) 241-256

Thus, B ( H ) is

P'o

a~ V~o,,

Stage 3. H B < H < Hc~ - A H (descending). This stage is the same as stage 3 of Section 4.1.1. except for the interval of H. B ( H ) is given by Eq. (34). Stage 4. H a < H < H c~ + A H (ascending). This stage is the same as stage 3 of Section 4.1.1. except for the interval of H. B ( H ) is given by Eq. (34). Stage 5. Hcl + A H < H < H A (ascending). In this stage, the supercurrent J ( = - J c ) penetrates from the sample surface until x = x 5 and the effective field H~ef is Hefe (1). From Eq. (7), the local flux density bs(x) is expressed by

P'o

+ j.rX3'x~b2Bd( xx) XzB /'1"0 +f

a b3z ( x ) x X3 z

dx

~'~0

I

J

= - - G I ( q 2 B ) q- G 2 ( q 2 B ) q- Gl(q3z)

+ G2( q3z) + G3(q0 ) - G,( qo ).

(46)

Stage 8. Hc i + A H < H < H A (ascending). In this stage, the supercurrent J ( = - J c ) penetrates from the sample surface until x = x 4 and the effective field Hae is Haf (1). From Eq. (7), the local flux density b4(x) is expressed by b4(x)

I~o

bs( x ) /x o

= "Px+H-HcI

- AH-Hp

a

( x 5 < x < a). (50) Using the boundary condition at x = x 5, bs(x 5) = b I z(Xs), we obtain 2Hp- H + H~I + A H xs = a. (51)

2Hp

= Hpx+H-H~,-AH-Hp a

Thus, B ( H ) is

( X4 ~ X ~ a ) .

(47)

Using the boundary condition at x = x 4, b4(x 4) = baa(X4), we obtain H B-H+2Hcl X4 =

+2AH+2Hp

2Hp

a.

B(__H) = 2 I rXl~xb°A( X)

tZo

B ( H ) = __2

dx

.x5 b lz( x ) ~o dx

(48)

+ faxbs(x, "oX) dx) + Gx(qs) + Ga(q5) + G3(qo) - G4(q0).

= --G,(qlz) +Gz(q,z) x ,Bx

a2 ~, XOA

dx /Z0

x - - d x x2B t~o

Same as Eq. (27).

+ 'faxbg(ixox) x, dx)

= --G,(q2a) + Ga(q2a ) + G,(q4) + G2(q4 ) + G3(qo ) - Ga(qo ).

(49)

4.1.2. For - H e 1 - A H < H a < He1 - A H Stage 1. H A - 2 A H < H ~ H A (descending).

Same as Eq. (27). Stage 2. Hcl -- A H < H < H a - 2 A H (descend-

ing). Same as Eq. (30).

(52)

4.1.3. For Hcl - A H < H B Stage 1. H A - 2 A H < H < H A (descending).

X, b2B ( X )

+f

-~ ~ Xo,, tZo + Jx,~ x

Thus, B ( H ) is

/J'0

247

Stage 2. H a < H < H A - 2 A H (descending). This stage is the same as stage 2 of Section 4.1.1. except for the interval of H. B ( H ) is given by Eq. (30). Stage 3. H B < H < H B + 2 A H (ascending). In this stage, the flux is completely trapped. The supercurrent J ( = Jc) pe/aetrates from the sample surface until x - ~ x l a and the effective field Herf is Heff (24). From Eq. (7), the local flux density bib (x) is expressed by biB(X) /x 0

H p x + HB

Hcj+AH+Hp

a

( X,B < x < a).

(53)

248

S. Tochihara et al. / Physica C 268 (1996) 241-256

Using the boundary condition at x = xtB, blB(XIB) ~- boA(XIB) , we obtain HB-HA

+ 2AH + 2H p

XIB =

a.

2Hp

(54)

]ZO

4.2.1. For a B <~ - a c l

2 (f[,.

a2

oa

+

B(H)/Ixo=HA-Hcl-AH-~Hp. boa(X ) x - -

dx

B(H)IIz o = --Gl(ql ) + G2(q,)

~o

= --G,(qIB)+G2(q~B)

+ H A - He, - A H - ½Hp.

4- G3(qo) - G4( qo)"

(55)

Stage4. H B + 2 A H < H < H A (ascending). In this stage, the supercurrent J ( = - J c ) penetrates from the sample surface until x = x 6 and the effective field Haf is Heee (1). From Eq. (7), the local flux density b6(x) is expressed by b6(x ) Hp - x + H - Hcl - A H tXo a

Hp

(x 6
a).

(56)

Using the boundary condition at x = x 6, b6(x 6) = biB(X6), we obtain H B -H+2AH+2Hp 2 Hp

a.

(57)

8(H) /z0

Gl( qlz) + G2( q l z )

+ HA- Hc,-AH-~Hp.

(61)

Stage 4. --Hcl -- A H + H~-~ < H < -He1 - A H (descending). H~m is the reverse full-penetration field (on the descending branch) for the medium-H A case, which is represented schematically in Fig. 2a. From the figure, we find H~rm = H + H e I + A H. Therefore, H~rm can be determined by taking x 2 = 0 in Eq. (36): H~rm=HA-Hc,-

AH-

2Hp,

(62)

B(H)

a22( f f ' " b --IZo °A( x6 bib ( x ) +f x xm /ZO

(60)

Stage 3. - H c i - A H < H < He i - A H (descending):

--=. /x0

Thus, B ( H ) is B( H)txo

(59)

This equation is the same as Eq. (18) with H = H A. Stage 2. H¢t - A H < H < H A - 2 A H (descending):

dx

/'to

x s

x6 =

- A H 4- Hprm

Stage 1. H A - 2 A H < H < H A (descending):

Thus, B ( H ) is B( H )

tion, we give only the final B ( H ) equations derived by a similar process as described above.

- G l ( q 2 ) 4- G2(q2) + H A - H¢I - A H -- 3Hp.

dx

Stages 5. H B < H < scending): dx

(de-

8(H) H + Hc, + A H + ~H o.

(64)

/Zo

+~x~["xb6(lxoX) dx)

Stage 6. H B < H < H B + 2 A H (ascending): B(H)

= - - G I ( q l B ) 4- G2(qlB) 4- G l ( q 6 ) 4- G2(q6 ) 4- G3(qo ) - G4(qo ).

-HcI-AH+H~m

(63)

(58)

4.2. Hysteresis loops for the medium-H a case (Hcl + a H + Hp < H A < Hc, + A H + 2Hp) In the intermediate case, there are four cases depending on the magnitude of H B. To avoid repeti-

' (65) H B +Hez + A H + -~Hp. /Xo Stage 7. H B + 2 A H < H < --Hel + A H (ascending):

8(H) _ _ /Xo

= G,(q3 ) + G2(q3) ' + H s +Hc~ + A H + 3Hp.

(66)

249

S. Tochihara et a l . / Physica C 268 (1996) 241-256

10. Hc~ + A H + H ~ m < H < H A (ascend-

Stage

H etf

ing):

Ca)

2Hp

:£

B(H)/tZo

= H = -ncl

-

-

A H - -'~ n p .

(70)

HA - H c~ - AH Hp 4.2.2. F o r - H c j Stage

X a

1.

- A H + H~-~m < H B < - H c l

HA - 2AH < H < HA

Same as Eq. (59). S t a g e 2. H c l - A H < H < H A - 2 A H ing). Same as Eq. (60).

- AH

(descending). (descend-

S t a g e 3. - H c i - A H < H < H¢ 1 - A H (descend-

ing). Same as Eq. (61). S t a g e 4. H B < H < - H c i - A H (descending). This stage is the same as stage 4 of Section 4.2.1. except for the interval of H . B ( H ) is given by Eq. (63). S t a g e 5. H B < H < H B + 2 A H (ascending):

H eff

(b) H+ prm

a

8(H)

X

_ _ /-to

= - G , ( q 2 B ) + G2(q2B ) all- H A -- Hcl -- a n

H a + H , +AH Fig. 2. Schematic representations of the reverse full-penetration fields (a) H~rm in the descending branch where Heft is Hal (3) defined by Eq. (21c) and (b) Hp+m in the ascending branch where Hal is Hen (1) defined by Eq. (21a). H A and H B are the maximum and minimum values of H, respectively.

- Inp.

(71)

S t a g e 6. H a + 2 A H < H < - H c i + A H (ascend-

ing):

B(H) /-to

= - - G I ( q 2 B ) + G2(q2B ) + G l ( q 3 ) + G2(q3)

S t a g e 8. - Hc 1 + A H < H < H c I + A H (ascend-

ing):

+ H A - Hcl - A H - -1f l i p .

(72)

B(H) tXo

= Gl(q3z) +GE(q3z)

S t a g e 7. - H c i + A H < H < H e i + A H (ascend-

ing): 1

+H B+Hcl+AH+-gHp.

(67)

Hc,

+ A H < H < H¢I + A H + H ~ ' r m (ascending). H~r m is the reverse full-penetration field Stage

9.

(on the ascending branch) for the medium-H A case, which is represented schematically in Fig. 2b. From the figure, we find H ~ m - - H - H c l - A H. Therefore, H ; m C a n be determined by taking x 4 = 0 in Eq. (48): H ; m = H a + Hci + A H + 2Hp,

(68)

B(H) _ _ /Xo

= Gl(q4 ) + G 2 ( q , ) + H B + Hcl + A H + ~l n p .

+ G2(q2B )

+ G , ( q3z) + G z ( q3z)

+ H A - HcI - A H -

T! H p.

(73)

S t a g e 8. He1 + A ' H < H < H A (ascending):

B(H) /Xo

- /z0

= -G,(qzB)

= - - G t ( q 2 B ) + GE(q2~ ) -t- G,(q4 ) + G2(q4 )

(69)

!

+ H A - Hcl -- A H - 3 n p .

(74)

250

S. Tochihara et al. / P hysica C 268 (1996) 241-256

4.2.3. For - H c I - A H < H a <_Hcl - A H Stage 1. H A - 2 A H < H < H A (descending).

Hell

HA -H:I-AH

~

Same as Eq. (59). Stage 2. H c ~ - A H < H < H A - 2 A H (descend-

2Hp

ing). Same as Eq. (60). Stage 3. H a < H < H c l - A H (descending). This stage is the same as stage 3 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (61). Stage 4. H a <_H < H c i + A H (ascending). This stage is the same as stage 3 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (61). Stage 5. H c i + A H < H < H A (ascending):

H -p~ X

a

H elf

B(H) -G,(q,z)

+G2(qlz)

H ++

tXo

prh

+ G , ( q s ) + G2(qs)

+H A

-

H¢I

-

AH-

:£ (75)

3Hp.

He 4.2.4. For Hc~ - A H < H B Stage 1. H A - 2 A H < H < H A (descending).

a

Stage 2. H a < H < H A - 2 A H (descending). This stage is the same as stage 2 of Section 4.2. I. except for the interval of H. B ( H ) is given by Eq. (60). Stage 3. H a < H < H B + 2 A H (ascending):

--=--Gt(qIB)

+G2(q,B) l

(76)

+ H A - Hc, - A H + 7kip. Stage 4. H a + 2A H ,~ H < H A (ascending):

-

AH

X

H p~ +

Same as Eq. (59).

B(H)

- H:I

H e +H~I + AH Fig. 3. Schematic representations of the reverse full-penetration fields (a) H~rh in the descending branch where Herf is Heff (2) defined by Eq. (21b) and (b) H~rh in the ascending branch where Haf is Herf (4) defined by Eq. (21d) and H;r ~- in the ascending branch where Herf is Heff (1) defined by Eq. (21a). When Heff is equal either to Hp-rh or Hp+rh or Hpr+h+ , B~(x) never crosses over the x axis.

B(H) _ _ /x0

= --G,(qlB) + G 2 ( q , B ) 4- a l ( q 6 ) 4- G 2 ( q 6 ) 1

+ H A -

Hcl - A H - 7Hp.

(77)

4.3. Hysteresis loops f o r the high-Ha case (Her + A H + 2Hp < H A )

There are five cases depending on the magnitude of H a. Only the final B ( H ) equations are given. 4.3.1. For H 8 < - H c l - A H - 2Hp Stage 1. HA - 2 A H < H < H A

Same as Eq. (59).

(descending).

Stage 2. Hc~ - A H + H~r~ < H < H A - 2 A H (descending). H~h is the reverse full-penetration field (on the descending branch) for the high-HA case, which is represented schematically in Fig. 3a. From the figure, we find H~h = H - Hcl + A H. Therefore, H~h can be determined by taking x I = 0 in Eq. (29): H ~ . = HA -- Hc~ -- a H

- 2Hp.

(78)

This stage is the same as stage 2 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (60).

251

S. Tochihara et al. / Physica C 268 (1996) 241-256 S t a g e 3. n c l -- A H < H < H c l

-

AH +Hpr h (de-

scending): l B( H)/tZo = H - Hcl + A n + ~Hp.

(79)

S t a g e 4. - H c i - A H < H < H c i - A H

(descend-

ing):

8( H)/

(80)

o = np.

S t a g e 5. H a <_ H < - HcI - A H (descending). This stage is same as stage 5 of Section 4.2.1. except for the interval of H . B ( H ) is given by Eq. (64). S t a g e 6. H a <_ H < H a + 2 A H (ascending). Same as Eq. (65). Stage

7.

HB+2AH
+AH+H~h

(ascending). H~h is the reverse full-penetration field (on the ascending branch) for the high-HA case, which is represented schematically in Fig. 3b. From the figure, we find H~rh = H + H c l - A H. Therefore, H~h can be determined by taking x 3 = 0 in Eq.

S t a g e 5. H B < H < - H c i - A H (descending). This stage is the same as stage 5 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (64). S t a g e 6. H B < H < H B + 2AH (ascending). Same as Eq. (65). S t a g e 7. H B + 2 A H < H < - H c i --t- m H (ascending). Same as Eq. (66). S t a g e 8. - H c i + A H < H < H c i + A H (ascending). Same as Eq. (67). S t a g e 9. Hcl + A H < H < Hcl + A H + H ~ r h (ascending). This stage is the same as stage 9 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (69). S t a g e 10. Hcl + A H + H~rh < H < H A (ascending). This stage is the same as stage 10 of Section 4.2.1. except for the interval of H . B ( H ) is given by Eq. (70).

(42): H~rh=H B

+Hcl + A H + 2Hp.

(81)

This stage is the same as stage 7 of Section 4.2.1. except for the interval of H . B ( H ) is given by Eq. (66). S t a g e 8. - H c I + A H +H~rh < H < - H c l

+ AH

(ascending): 8 ( n)/IXo = n + nc~ - A n -

1 ~np.

(82)

S t a g e 9. - H c~ + A H < H < H cI + A H

(ascend-

ing): B( H)/IXo

=

(83)

- - ~'H p .

S t a g e 10. H ~ + A H ~ H ~ H A (ascending). This stage is the same as stage 10 of Section 4.2.1. except for the interval of H . B ( H ) is given by Eq. (70). 4.3.2. F o r - H c j - ZlH - 2Hp < H a < S t a g e 1. H A - 2AH < H < H A

-ncj

-

AH

(descending).

Same as Eq. (59). (descending). This stage is the same as stage 2 of Section 4.2.1. except for the interval of H . B ( H ) is given by Eq. (60). S t a g e 3. H c i - A H < H < H c ~ - A H + Hpr h (descending). Same as Eq. (79). S t a g e 4. - H c l - A H < H < H~l -- A H (descending). Same as Eq. (80). S t a g e 2. H c i - A H + H~rh < H < H A - 2 A H

4.3.3. F o r - - n c l - - z l H < H B < Hcl - z l H S t a g e 1. H A - 2AH < H < H A (descending).

Same as Eq. (59). S t a g e 2. Hc~ - A H + H~r h < H < H A - 2AH (descending). This stage is the same as stage 2 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (60). S t a g e 3. H c i - A H < H <_ H c l - - A H + Hprh (descending). Same as Eq. (79). S t a g e 4. H a < H < Hcl -- A H (descending). This stage is the same as stage 4 of Section 4.3.1. except for the interval of H. B ( H ) is given by Eq. (80). S t a g e 5. H a <_ H < H c i + A H (ascending). This stage is the same as stage 4 of Section 4.3.1. except for the interval of H. B ( H ) is given by Eq. (80). S t a g e 6. Hcl + A H < H < Hc~ + A H + 2 H p (ascending): B(H)/txo=G,(qs)

+ G2(q5 ) +½Hp.

(84)

S t a g e 7. Hcl + A H + 2 Hp < H < H A (ascending). This stage is the same as stage 10 of Section 4.2.1 except for the interval of H. B ( H ) is given by Eq.

(70). 4.3.4. F o r Hcl - A H < H B < Hcl + A H + Hprh S t a g e 1. H A - 2AH_< H < H A (descending).

Same as Eq. (59).

252

& Tochihara et al./ Physica C 268 (1996) 241-256

Stage 2. Hc~ - A H + Hprh < H < H A - 2 A H (descending). This stage is the same as stage 2 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (60). Stage 3. H a <_H <_H cI - - A H + Hp~h (descending). This stage is the same as stage 3 of Section 4.3.1. except for the interval of H. B ( H ) is given by Eq. (79). Stage 4. H B <_H < H B + 2 A H (ascending). B ( H ) / t z o = H B - H c , + A H +-~Hp.

(85)

Stage 5. H B + 2 A H < H < Hc, + A H + H ; + ~ (ascending). Hp+~- is the reverse full-penetration field (on the ascending branch) for the high-HA case, which is represented schematically in Fig. 3b. From the figure, we find Hp+r~- - ( H B - H ¢ I - A H ) = 2Hp. We obtain

H;+2 = H B - Hc, - A n + 2Hp,

(86)

B( H)/tx o = 01(q6) + 62(q6) + HB-

He1 +

An +½np.

(87)

Stage 6. H¢l + A H + Hp+~2 < H < H A (ascending). This stage is the same as stage 10 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (70). 4.3.5. For H~I + AH + Hprh ___H a < H A Stage 1. H A - 2AH_< H_< H A (descending). Same as Eq. (59). Stage 2. H a < H <_H A - 2 A H (descending). This stage is the same as stage 2 of Section 4.2.1. except for the interval of H. B ( H ) is given by Eq. (60). Stage 3. H a <_H <_H B -b 2 A H (ascending): B ( H ) / I ~ o = --Gl(qlB) + G2(q,B) -[-.HA - H c l

-AN

- 3Hp.

(88)

Stage 4. H a + 2 A H _< H _< H A (ascending):

e(H) _ _ /Zo

-- G l ( q l B ) + G2(q,B )

+Gl(q6) + G2(q6) +HA - H e , - An-- Hp.

(89)

5. C o m p u t e d

M(H)

curves

We have analytically confirmed, for each case in Section 4, that all the B ( H ) equations derived at each stage are continuous at their end points. Using these B ( H ) equations in Eq. (9), we can draw the computed M ( H ) curves under any experimental condition of applied fields. To carry out the evaluation, one period of the alternating magnetic field (0_< w t < 27r) was divided into 360 regular intervals, and M(tot) was numerically obtained at each point. In this section, we give some computed M ( H ) curves, where both the axes are normalized to the full penetration field Hp. Figs. 4 - 6 give the initial and hysteresis M ( H ) curves for three different dc bias fields, H0c = 0, Hp and 4Hp; and for each H0c, four different sets of (a) A H = 0, Hc~ = 0, (b) A H = 0.1Hp, Hcl = 0, (C) A H = 0, H¢1 = H p / 3 and (d) A H = 0 . 1 H p , H c l = H p / 3 . For each case, three M ( H ) loops are drawn for Hac = H p / 2 , 2Hp and 4Hp. From Figs. 4-6, we note the following aspects: (a) As seen in Figs. 4a, 5a and 6a, the M ( H ) loops A H = Hc~ = 0 show a typical profile of the Bean model and shift to the right by H / H p = 1 in Fig. 5a ( H d c = H p ) and by H / H p = 4 in Fig. 6a (ndc = 4Hp). (b) When A H ~ 0, the M ( H ) loops are expanded up and down with the increase of A H, but the profile is essentially the same as those for A H = 0. This means that the surface barrier is not explicitly reflected on the shape of the M ( H ) curves, and practically it is difficult at a glance to distinguish the existence of A H from the observed profile of the M ( H ) loops. However, more detailed examination of the M ( H ) curves may reveal the existence of nonzero values of A H. For example, we compare Figs. 4a and b, where A H = 0 and 0.1 Hp, respectively. When the applied field is reversed, the magnetization drops at the Meissner slope until the reduced fraction of M ( H ) becomes 2AH. As shown in Fig. 7a ( A H = 0; expanded figure of Fig. 4a), the M ( H ) curve reversed at H / H p = 4 does not trace the Meissner slope given by a straight line segment, but draws a curve. On the contrary, as seen in Fig. 7b ( A H = 0.1Hp; expanded figure of Fig. 4b), the M ( H ) curve reversed at H/Hp = 4 traces well the Meissner slope

S. Tochihara et al. / Physica C 268 (I996) 241-256

until 2AH/Hp=0.2 and then begins to draw a curve, suggesting that the M(H) profile around the reversed point could be a good probe for the surface barrier. (c) When Hcl :# 0, the M(H) loops change the shape drastically. As seen in Figs. 4c and 5c, the loops show an apparent step-like feature. Besides, for Hdc = 0, the loop is still symmetrical about the

253

. . . .


1

\ ½\\ \

~0

-1 -4

-2

0

H/Hp

2

4

(b)

1 i

~

t

I

i

\\\\\

(a)

\ \\\

-1 -4

I

-4

.

I

t

I

I

-2

0

2

4

t

•

~

I

I

2

4

J

i

i

.

i

(c)

1 '

i

.

i

(b)

1

--.

I

0

H/Hp '

H/Hp i

I

-2

\

0

\

-1

-1

-Z I

I

I

I

I

-4

-2

0 H/Hp

2

4

i

I

I

I

-2

0

2

4

i

H/Hp

.

i

i

1

(d)

(c)

1

~o 0

-1 t

-4 I

I

-4

-2

I

I

I

0

2

4

H/Hp i

J

i

i

.

i

(d)

1

-2

I

I

0 2 H/Hp

I

4

6

Fig. 5. Theoretical M(H) curves, scales by Hp, for Hd¢ = HI,; (a) A H = 0 , Hcl = 0 , ( b ) A n = 0.1Hp, Hct = 0 , ( c ) A H = 0 , Hcl = np/3 and (d) An=0.1Hp, nc] = rip/3. In each figure, loops are shown for Hac = H p / 2 (smallest), 2Hp and 4Hp (largest).

~o -1 I

I

I

I

I

-4

-2

0 H/Ho

2

4

Fig. 4. Theoretical M(H) curves, scales by Hp, for Hoe = 0; (a) A H = 0 , Hot = 0, (b) A n = 0.1Hp, Hcl = 0, (c) A H = 0 , Hcl = Hp/3 and (d) A n = 0.1Hp, He] = Hp/3. In each figure, loops are shown for Hac = Hp/2 (smallest), 2Hp and 4Hp (largest). Note that the smallest loop in (c) and (d) is overlapped on the initial magnetization curve.

origin (Fig. 4c), while for Hdc = Hp, the asymmetric nature of the loop becomes evident (Fig. 5c). When Hoe is large enough, we find traces of this asymmetric nature at the top comer of the left-hand side in Fig. 6c. Such a profile with a distinctive feature easily permit us to recognize the role of H~I. (d) When A H ~ 0 and Hcj ~ 0, the M(H) loops show essentially the same behavior as those shown in Figs. 4c, 5c and 6c, but are somehow expanded up

254

S. Tochihara et al./ Physica C 268 (1996) 241-256

and down due to the existence of A H, as demonstrated in Figs. 4(t, 5d and 6d. To see the role of A H and H~] from another viewpoint, we give several sets of the loops using A H and H~z as parameters. In Fig. 8, we show the case for (a) Hdc = 0 and (b) Hdc = Hp, where Hcl is fixed at Hp/3. Since Hcl is not zero, each loop has a step-like feature as mentioned above. Each of Figs.

0.1

•

I

I

'

I

'

I

0

'

a)

a -0.1

7-

-0.2 -0.3 -0.4 3.6

' 3.7

' ...... 3.8 3.9

4

4.1

H/Hp !

i

i

i

I

(a)

1

\\\

o.

7" ~..0

0 -0.1

\

I

'

I

'

I

.

,

I

'

(b)

-0.2 :~ -0.3

I

t

0

2

[

[

I

4

6

8

-0.4

2AH / Hp

H/Hp i

"

i

*

i

,

i

,

-0.5 3.6

i

(b)

1

, 3.7

,.,. 3.8 3.9 H/Hp

t , 4 4.1

o.

~o

Fig. 7. Expanded figures of (a) Fig. 4a (for A H = 0) and (b) Fig. 4b (for 2AH/Hp=0.2). Straight line segments indicate the Meissner slope.

-1 I

I

I

I

I

0

2

4 H/Hp

6

8

i

i

,

i

,

J

,

J

(c)

1 o_

~0 -1 I

I

I

1

I

0

2

4

6

8

H/Hp i

•

i

,

i

,

i

1

(d)

1 a.

~0

\\\ \

8a and b includes three loops which correspond to A H = 0 (small dots), 0.2 Hp (open circles) and 0.4 Hp (solid circles). As seen in the figures, the loops are expanded up and down as A H increases, but the essential profile does not change even for different values of AH. In Fig. 9, we show the case for (a) Hdc = 0 and (b) Hdc = Hp, where A H is fixed at 0.1 Hp. In each figure, three loops are drawn; H~] = 0 (small dots), 0.2H o (open circles) and 0.4Hp (solid circles). For Hcl = 0, the M(H) loops do not show any step-like feature. However, for Hcl 4~ 0, the loop contains the step and it becomes larger with the increase of Hcl.

-1 I

0

I

2

4 H/Hp

I

I

6

8

Fig. 6. Theoretical M ( H ) curves, scales by Hp, for Hac = 4Hp;

(a) A H = 0 , H¢]=O, (b) A H = 0 . 1 H p , H c l = 0 , (c) A H = 0 , Hcl = Hp/3 and (d) AH=O.IHp, Hcl = Hp/3. In each figure, loops are shown for Hac = H p / 2 (srnalles0, 2Hp and 4Hp (larges0.

6. S u m m a r y

In the framework of the modified Bean model which takes into consideration explicitly the surface barrier A H and a nonzero value of the lower critical field Hcl, we have analytically derived magnetiza-

S. Tochihara et a l . / Physica C 268 (1996) 241-256

tion equations for type-II superconductors immersed in an alternating magnetic field superimposed on a dc field, H = H d c +Hac cos(cot), where H o c > 0 and H~ > 0. We consider an infinite cylindrical specimen and the external field is applied along the axis. Denoting the maximum and minimum values of H by HA( = Hdc + Hac) and HB(= ndc -- nac), respectively, four types of hysteresis loops appear, depending on the magnitude of H A. Each of these is further classified into several cases, depending on the magnitude of H B. The magnetization equations which construct full hysteresis loops were derived by decreasing H from H A to H a , forming the descending branch, and then by increasing H from H a to H A, forming the ascending branch. To bring full hysteresis loops to completion, one needs to derived the magnetization equations of 83 stages of H and all the equations are thoroughly presented. To verify our derivations, all

I

(a)

-1 I

I

I

-2

0

2

i

4

H Hp I

I

I

I

(b)

1

£o -1 i

I

0

i

H/Hp

I

I

2

4

Fig. 9. Theoretical M(H) curves, scales by Hp, for (a) Hdc = 0 and (b) HOe = Hp, where A H = 0.1 Hp. Ill each figure, loops are shown for Hcl = 0 (small dots), 0.2Hp (open circles) and 0.4Hp (solid circles). Some dots or circles are omitted for clarity.

I

(a)

1

I

£o

I

I

I

1

-2

1

255

£o -1 -2

0 H Hp

I

2

I

1

4

I

(b)

-1 I

I

I

I

-2

0

2

4

H/Hp

Fig. 8. Theoretical M(H) curves, scales by Hp, for (a) Hdc= 0 and (b) //de= Hp, where Hcl = Hp/3. In each figure, loops are shown for AH= 0 (small dots), 0.2Hp (open circles) and 0.4Hp (solid circles). Somedots or circles are omittedfor clarity.

the equations were confirmed to be continuous at their end points. Further test of the present derivations was made by comparing them with the available magnetization equations so far reported. For example, Eqs. (30) and (37) (descending) and Eqs. (43) and (49) (ascending) were confirmed to agree with Eq. (5) in Ref. [2]. Using the magnetization equations derived here, we computed the M(H) curves for various experimental conditions. Some typical results are demonstrated in Figs. 4-9. We found that the surface barrier A H does not essentially deform the M(H) loops, but they are merely expanded up and down with increasing A H. On the contrary, the shape of the M(H) curves is drastically changed by a nonzero value of Hcl, i.e., the loops show a step-like feature. Model calculations presented here have plainly revealed the effect of A H and H d on the hysteresis loops of type-II superconductors. Using the K i m -

256

S. Tochihara et aL / Physica C 268 (1996) 241-256

Anderson model modified by taking into consideration A H and Hc~, more elaborate magnetization equations are expected to be obtained, although analytical treatments should be much more complicated.

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[12] [13]

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