Z. Physik B 30, 367-373 (1978) Zeitschrift for Physik B © by Springer-Verlag 1978

Spin Wave Excitations in the Dilute Two-Dimensional Ferromagnet K2Cu _ xZnxF4

V. Wagner* **

Institut Max yon Laue-Paul Langevin, Grenoble, France

U. Krey

Fachbereich Physik der Universit~it Regensburg, Germany

Received April 21, 1978

The magnetic excitations of the dilute two-dimensional ferromagnetic Heisenberg system K2Cu 1 _xZnxF4 have been studied by inelastic neutron scattering for x = 0, 0.08 and 0.22, at T= 2 K. The dispersion of the frequency and of the linewidth has been obtained for wavevectors q in the direction of the strongest dispersion, q~(1, 1,0), within the plane containing the strong exchange interaction. In the dilute samples, x = 0.08 and 0.22, the effect of the nonmagnetic impurities on both the frequency and the linewidth has been observed. Excellent agreement is obtained between the experimental results and the predictions of a continued-fraction-computer-simulation of the lineshapes, for the sample with x = 0.08. For x = 0.22, there is still good agreement, although the experimental frequencies are slightly higher than predicted.

1. Introduction

Presently the excitations of dilute magnetic systems are studied extensively, both by experiment and theory. However, hithertoo, as far as the authors know, experi- ments have been concentrating exclusively on anti- ferromagnets. (For a short review see [,,1].) Especially interesting are the so-called quasi-bidimensional magnets, bulk crystals with ferromagnetic or anti- ferromagnetic planes, which are almost decoupled: For these systems, the effects of disorder should be more pronounced than in systems with three-dimen- sional magnetic behaviour. In fact, the most recent, and most thorough studies have been performed on the system Rb2Mn a _xMgxF4 [2]. This system shows quasi-bidimensional antifer- romagnetic behaviour with s = 5/2 for the Mn ions, while the system studied presently is a quasi-bidimen- sional ferromagnet, with s= l/2 for the Cu ions. The interest concerning such systems is concentrating mainly on three points:

(i) the critical behaviour, i.e. critical exponents and

* Guest scientist from Phys. Institut der Universit~it, D-8700 Wiirzburg, Germany ** Now at Phys. Technische Bundesanstalt, Bundesallee 100, D- 3300 Braunschweig, Germany

universal lineshape functions near the phase transition, T~ T~, see [-3] ; (ii) the behaviour of the magnetic correlation functions near the so-called percolation concentration xc, which is defined as the minimum dilution x which is necessary to destroy the magnetic long range order (T~ ~ 0), see [4]; (iii) the low-temperature magnetic excitation spec- trum, i.e. dispersion relations, linewidths, and line- shapes as a function of x, see [-2].

For all these different points there exists a lot of theoretical work, allowing for definite predictions, which should be tested experimentally (see e.g. [-5- 9, 10-213, and [-1, 22-24], for the points (i), (ii), and (iii), respectively). In particular, concerning the low-temper- ature excitation spectrum of dilute antiferromagnets, it has been predicted theoretically, [22, 24 27], that the scattering intensity S(q, co), when presented as a func- tion of the energy hco transferred from the neutron to the spin system, for given momentum change h q, should show a pronounced substructure, reflecting the different Ising energies of magnetic ions in different environments. In [-2] this substructure has actually been found for the

0340-224X/78/0030/0367/$01.40

368 V. Wagner and U. Krey: Spin Wave Excitations

quasi-bidimensional Rb2Mnl_~MgxF4 system, as it had before in the three-dimensional antiferromagnet Mno.32Zno.6sF 2 [28]. 0 n the other hand, concerning the point (ii), the most interesting result was the observation of one-dimensional paths connecting mag- netic sites for concentrations near the percolation limit x c = 0.41 in RbaMn 1 _xMg~F4, see [4]. One might ask, therefore, whether similar effects would also appear in ferromagnetic quasi-bidimensional mag- nets, or whether, due to the different ground state and different interactions, there would be substantial differ- ences compared with the antiferromagnetic case. The present paper is concerned with this question, for the case of the low-temperature excitations (point (iii)). The system considered is the quasi-bidimensional dilute ferromagnet K2Cu I _xZnxF 4. The scattering law S(q, (~) for inelastic neutron scattering has been studied experimentally for x = 0 , 0.08, and 0.22, and also calculated by a computer simulation for the same dilutions, and additionally for the limit case of percola- tion, x = x c = 0.41, where the magnetic long range order is destroyed in a square lattice. From the computer simulation, which is based on a continued fraction algorithm described elsewhere [29], one concludes that in ferromagnetic dilute systems, too, there should be a substructure in the lineshapes. This has also been predicted by R. Alben from computer simulations using a different technique [30]. However, this substructure, which has also been found in calcu- lations for three-dimensional ferromagnets [31], is on a much finer scale than for the antiferromagnetic systems, and it could not be seen by the present experiment. But the effect of nonmagnetic impurities on the dispersion curves for frequency and linewidth could be observed: For the crystal with x = 0.08, excellent agreement was obtained between experiment and theory; for x = 0.22, there is still good agreement, although the experimental frequencies are slightly higher than predicted.

2. Experimental

2.1. Samples

The samples were single crystals of typically 1.5 cm in diameter and 3.0 cm in length. They were grown by a Czochralski technique at Cristaltec (Grenoble). Three samples with Zn-concentrations x = 0 , 0.08, and 0.22 were available. The pure sample, x = 0, was an as-grown crystal, while the dilute samples were cleaved from larger twinned crystals. The mosaicity increased with the Zn-concentration and was 10', 30' and 80', re- spectively. The Zn-concentration was determined by chemical analysis, and no evidence for a concentration gradient was found.

Table 1. The lattice constants a and c for the three crystal samples of K2Cu I ~Zn~F~ are presented, as measured by neutron Bragg scattering at T=2 K

x = 0 x = 0.08 x = 0.22

a 4.119 ,~ 4.107 A 4.095 _+_ 0.005 _+ 0.005 _+ 0.005

c 12.678 A 12.680 A 12.765 A _+ 0.005 _+ 0.005 _+ 0.005

In the following, we refer all quantities to the tetragonal Bravais lattice which describes the spin arrangement, but neglects small Jahn-Teller distortions in the neigh- bourhood of the Cu ion. The lattice constants observed at 2 K from neutron Bragg scattering are given in Table 1. For x = 0, our results agree with those of [32], where also a representation of the unit cell can be found. Upon doping with Zn, the tetragonal axis c increases, while the nearest-neighbour-distance a among Cu ions decreases. The relative changes are apparently nonlinear in x. For x =0.22 the relative decrease of a is A a/a ~- - 0 .5 per cent.

2.2. Measurements

The measurements were performed with the triple axis crystal spectrometer IN3 on a thermal neutron guide of the High Flux Beam Reactor of the Institute Laue- Langevon. A composite Cu(111) crystal with a radius of vertical curvature Rv= 1.9m was used as monochro- mator, and a fiat Zn(002) crystal as analyser. Typical horizontal collimation was 30-40-40-70rain. Most data were taken keeping the wavevector of the scattered neutrons fixed at k~=2.3,~ -1, while a few data were taken at fixed incident energy with a graphite filter at k i = 2.673 A- 1 For the inelastic measurements the samples were held at 2.2 K by a I-Ie-cryostat. For the doped samples, a bath cryostat had to be used, a fact which unfortunately increased the background. The samples were oriented with a [-1i0] axis perpendic- ular to the scattering plane. The inelastic scattering cross section was measured at constant wavevector transfer with q = (re~a). (~, ~, 0) in the central Brillouin zone. In this "longitudinal" scattering geometry the effect of the mosaic spread on the linewidth is smallest. Typical observed neutron groups are shown in Fig- ure 1. In the case of pure K2CuF 4 the linewidth was given by the instrumental resolution, which is between 50 and 70GHz for the full width at half-heigth, in frequency v = co/2~ (compare Figure 4 below). The resolution obtained was sufficient to observe the effects of dilution both for x=0.08 and x=0.22. In

V. Wagner and U. Krey: Spin Wave Excitations 369

>--

Z Ld t-* Z

LLI >

I'-" < J LLI I:Z::

~ . X = O TIMES 27 X =0.08

TIMES 12NXj !

t J

X=0.22 !~I

÷o

~+++

1.0 1.5 ENERGY [THZ]

o

~x

2'.0 =

Fig. 1. The scattering intensity S(q, E), henceforth to be called magnon lineshape, is presented for inelastic scattering of neutrons from the ferromagnetic qnasi-bidimensional dilute system KzCul_xZnxF 4 with x = 0, 0.08, and 0.22, as a function of the energy transfer E, given in THz, that is, for v=E/h, for a wavenumber transfer of q = (~/a). ~. (1, 0, 0) in the plane of strong exchange, with ~ = 0.75; ~ = 1 would correspond to the Brillouin zone boundary. Temperature T =2.2K

order to obtain sufficient statistics, the counting times had to be increased as much as to 30 min per point for x = 0.22 because of the strong line broadening.

3. Calculations

To compare experiment and theory, we have performed computer simulations for a 80 x 80 square lattice of randomly dilute Heisenberg spins interacting by a nearest neighbour exchange interaction at zero temper- ature. The quantity calculated is the dynamic structure function ("scattering law")

1 S(q, co) = - - Im {<q*(E + i t - H ) - 7q>, (1)

7~

where Im means the imaginary part, e a positive infinitesimal, E = ha) the excitation energy, and H the spin Hamiltonian in the Spin Wave approximation, which is exact at low temperatures:

I-I= - ~ 4m{l l><ll- lm>(l l} (2) 1,rn

In (2), the sum is only over sites l and m, which are occupied by a magnetic atom; II) is the state obtained from complete alignment by flipping the spin at 1 (the magnetic quantum number s is 1/2 in the present case). Iq) is a plane wave state,

N I q ) = N -1/2 ~ exp(-iq'rz)l/>,

l = l

where once more the sum is only over the occupied sites. Within the calculation we used the value Jz,, ~ 0.2367 THz for the nearest neighbour exchange integral, which corresponds to the pure system [32, 33]. As is well known, for a given wave number transfer q, the function S(q, co) should be directly proportional to the scattering intensity as measured in a constant-q- scan, with co varying. Within the calculation of (1) we used a continued fraction algorithm, as suggested recently for amor- phous systems by one of the authors [29]. This algorithm was originally devised by Haydock et al. [34] for the calculation of bulk- and surface-densities of states &(E) for ideal, tight binding electrons. Concern- ing the present case of a disordered system, the essential and apparently new idea is to apply this same algorithm to the Hamiltonian (2), but with Iq>, instead of I/), as the "starting state" of the algorithm. Then, instead of gl(E), one obtains directly S(q, co). This continued fraction algorithm is extremely fast and accurate, still faster and simpler than Albert's powerfull method [26], which would involve a direct integration of the equations of motion for the Green's function and subsequent Fourier transformation. In principle, both methods are equivalent, however in certain cases the present method may also be somewhat more accurate, as has been found from a comparison of the fcc-densi ty of states in [-35], Figure 6 and [36], Figure 43, re- spectively. We have calculated 40 continued fraction coefficients a m and b n, either, before applying the standard analyti- cal continuation described in [34]. In a moment expansion, this would correspond to 80 correct mo- ments.

4. Results

In Figures 2 a - d the experimental lineshapes (back- ground intensity not yet substracted !) are shown for x-- 0.22, with q = (~/a). ~. (1, 1, 0), for various values of ~, where ~ = 1 corresponds to the edge of the Brillouin zone. The (1, 1,0) direction has been chosen, since along this direction the gradient focussing works best. We have also checked that there is no measurable dispersion for q ~(0, 0, 1), both in the pure and in the dilute systems, which reflects the quasi-bidimensional magnetic behaviour of the system [32, 33]. In Figure2, at low frequencies, the increase of the background intensity with decreasing frequency is possibly due to a second order scattering "contami- nation" from the liquid Helium, into which the sample was immersed. The inset in Figure 2a is the result of a "fitting" program where a Gaussian lineshape has been used, the background intensity being substracted to- gether with the fitting procedure.

370

L COUNTS

-1000

500

COUNTS 200

100 / \ I I I

03 ENERGY cTHOp

~~ ~ =0.35

1000

500

0 0'.2 014 016 ~ a ENERGY (THZ) b

V. Wagner and U. Krey: Spin Wave Excitations

;OUNTS

\ = 0.50

i

i

1.0 1.5 ENERGY (THZ)

l

COUNTS COUNTS ooo {; = 0.75 I l y

' ' ~ , j . , r , ~-

, ,,, (~= 1.00

0( 015 110 1'.5 " 1.1 1.65 2.2 e ENERGY [THZ] d ENERGY (THZ)

Fig. 2a~l. The maguon lineshapes for x=0.22 are presented for different wavenumbers ~, see Figure 1; - x - x - results at 2.2K; --o--o--background intensity obtained at higher temperatures, in case of Figure 2a; the inset of Figure 2a represents the corrected results after subtraction of the background intensity

Similar results, which are not presented, have also been obtained for the samples with x=0 .08 and x = 0 , respectively. Since in all cases there is no substructure in the lineshapes, within the statistics, the only infor- mation from these results concerns the changes of frequencies and linewidths with increasing dilution. These changes can most clearly be extracted e.g. from Figure 1, where the three lineshapes for ~=0.75 are presented, which are obtained with x = 0.22, 0.08, and x = 0, respectively. Obviously, the frequency is lowered with increasing x, and the l inewid th -which for x = 0 is equal to the experimental reso lu t ion- increases considerably. The same conclusions can be drawn from the results of the theoretical calculation, see Figure 3. Here we have used a finite value of ~ = 0.07 T H z for the quantity e appear- ing in (1); this would correspond to a Lorentzian

experimental resolution lineshape, with the full half- intensity linewidth & v = 5. For x--0.08, the lowering of the frequencies, as found by experiment and theory, is so small that it can hardly be distinguished in the dispersion curve; e.g. for -- 0.75, see Figures 1 and 3, the relative shift is only 2 %. Therefore, for x = 0.08, in Figure 4, only the dependence of the linewidth on x is shown, since here the effects of dilution are more drastic. The dashed curve represents the experimental resolution, as measured for x = 0 ; it varies somewhat with the wavevector ~ because of focussing effects: The largest value, & v _-__ 0.07 THz, has been used in the calculations presented in Figure 3 for x = 0.22. The mean value, however, would be smaller, namely & v~0.057 THz. This last value for the experi- mental resolution yields the "theoretical linewidths" for x -- 0.08, which are presented as crosses in Figure 4,

v. Wagner and U. Krey: Spin Wave Excitations 371

together with the experimental results for the same dilution x. Obviously, the agreement between experiment and theory is excellent. In particular it should be noted that there is no "fitting parameter" apart from the experi- mental resolution width e in the theory. In Figure 5, both the frequency, and also the linewidths are presented over the wavevectors ( for the case of x =0.22; here the horizontal collimation was relaxed (30-60-60-70) compared with the case of x=0.08; therefore, as already mentioned above, within the theoretical calculation we used a higher value, e = 0.07 THz for the Lorentzian experimental resolution width in this case. Nonetheless, however, both the experimental frequencies and also the linewidths of Figure 5 are larger than predicted from the theory. The reason for this discrepancy is not quite clear; for the frequency it could be explained partially by a,-~ 5 % to 10% increase of the exchange integral, which might result from the observed change of the lattice constant a. The linewidths, for which the discrepancy is larger, may also be influenced by thermal effects, which have been neglected in the calculation, and which should be more effective in the strongly diluted system, x = 0.22, than for the case of x = 0.08. An additional broadening may also be caused by statistical fluctuations of the exchange integrals around their mean value, as a function of the position. Such fluctuations could result from internal strains and the effects of different environ- ments etc., which are present in strongly diluted systems. It would of course be interesting to know the lineshapes for very high experimental resolution, that is, for small e: Therefore, in Figure6a, b, c results are presented which have been calculated with e=0 .012THz for x =0.22, and also for the limiting concentration of x = 0.41, which corresponds to the percolation dilution, in case of nearest-neighbour exchange. Obviously, there is considerable structure in the line- shape, in particular for x=0.41, where any spin is connected only to a finite number of other spins by nearest-neighbour exchange bonds, which means that here all states are strictly localized; however, compared with the antiferromagnetic case, see [22-27], this structure is on a rather fine scale und should be difficult to observe: F rom Figure 6b one would conclude that the most promising case, concerning the observability of the fine structure would be (=0 .75 or 1.00 for x =0.41. In Figure 6c, the results for ( = 1 are compared, which are calculated with different experimental re- solutions, namely with e = 0.012, 0.035 and 0.070 THz: Obviously, it would be desirable to obtain a resolution much better than e=0 .03THz, together with mixed crystals of smaller mosaicity; only then details of the fine structure might become visuable.

S (q,c~) I 0.5 0.75 i i

1 o.s ! '" /,"

, ,,~8%Zn,

'", .~.._.------22% Zn

0 2 4. 8 10 E 2s3 =

Fig. 3. The same quantity as in Figure 1, for x=0.08 (- -) and x =0,22 ( ) for ~=0.25, 0.5, 0.75, and 1, as calculated from the computer simulation with an "experimental" resolution of 0.07 THz (full linewidth at half intensity, Lorentzian line profile). E/2sJ= 1 corresponds to 0.2367 THz; Y ~ n.n. exchange; s = 1/2: spin quantum number; $ exact frequencies for x =0

% 0.1

0.05

0 0

iIo I I Iii

o

Fig. 4. Wave vector dependence of full half-intensity linewidth of magnons in the diluted quasi-bidimensional ferromagnet K2Cul_xZnxF ~ for x=0.08, l=experimental result; ©=continued fraction computer simulation; the dashed line represents the experi- mental resolution

v//TH

2

1 Dq2 ~/~//,~,~"I \,,,x

0

Fig. 5. Magnon dispersion curve for K 2 Ci ~ xZn. ,F4 with x = 0.22; the bars give the full half-intensity linewidth, q experimental results; ? continued fraction computer simulation. The dashed line gives the corresponding dispersion curve in pure K2CuF4; q = Oz/a). (. (1, 1, 0)

372 V. Wagner and U. Krey: Spin Wave Excitations

1.0 x =0.22 1

x =0.41

l ~=0.25 0.50 0.75 1.00 0.6

uo , ~ = 0.25

0.4[ \ 0.50 0.75

0.2

°o 2 i 6 °o Z 8 :

a E/2s-3 E / 2s'J --

T s(q,w)

04 '

0.2

0 4 8

e E/2s 3

~=1.00

~ . . . # . . . . .

Fig. 6a-c. Magnon lineshapes, as calculated with the continued fraction algorithm for a Lorentzian instrumental linewidth of 0.012 THz; E/2sJ = 1 would correspond to 0.2367 THz; Y~ n.n. exchange a x=0.22 b percolation dilution x=0.41 c x=0.41; comparison of results with three different experimental resolutions 8: - - e=0.012THz; - e=0.036THz, -... e=0.07THz. The results of Figure 6c have been obtained by averaging over four different random configurations

The fact that for ferromagnets, in contrast to anti- ferromagnets, there should be no large scale structure in the lineshapes had already been predicted in [24] by one of the authors; the explanation is simply that for ferromagnets the region in q-space, where the Ising approximation makes sense, is not at the same time a region of zero group velocity, as it would be in antiferromagnets.

5. Conclusions

We have studied the low-temperature magnetic exci- tations of the quasi-bidimensional dilute ferromagnet KzCUl_xZnxF 4, for x = 0 , 0.08, and 0.22, both by inelastic scattering of neutrons, and by a computer simulation. The influence of the dilution on frequencies and linewidths was analyzed for wavenumbers trans- fers q in the (1, 1, 0) direction, within the plane of strong

exchange interaction. For x=0.08, there is excellent agreement between experiment and theory, while for x =0.22 both the frequencies and the linewidths are somewhat larger than predicted, if one assumes that the exchange integrals do not change with x. In contrast to the case of antiferromagnets there is no large-scale substructure in the experimental lineshapes. In fact, the substructure found theoretically is on a scale, which is smaller than the experimental re- solution available at present. We hope to obtain a crystal with dilution near the percolation limit in the near future; this would allow us to study also the critical behaviour and the percolation properties of the dilute quasi-bidimensional ferromag- net.

One of the authors (V.W.) would like to thank R. Alben for a stimulating discussion and for the communication of his unpublished results.

V. Wagner and U. Krey: Spin Wave Excitations 373

References

1. Cowley, R.A.: AIP Conf. Proceedings 29, 243 (1976), and references therein

2. Cowley, R.A., Shirane, G., Birgeneau, R.J. Guggenheim, H.J.: Phys. Rev. B15, 4292 (1977)

3. Birgeneau, R.J., Ats-Nielsen, J., Shirane, G.: Phys. Rev. B16, 280 (1977)

4. Birgeneau, R.J., Cowley, R.A., Shirane, G., Guggenheim, H.J. : Phys. Rev. Lett. 37, 940 (1976)

5. Grinstein, G., Luther, A.: Phys. Rev. B13, 1329 (1976) 6. Lubensky, T.C.: Phys. Rev. Bll , 3580 (1975) 7. Krey, U.: Physica 80B, 231 (1975), 82B, 349 (1976) 8. Suzuki, M.: J. Phys. C: Solid State Phys. 8, 255 (1975) 9. Krey, U.: Z. Physik B26, 355 (1976); B27, 325 (1977)

10. Stanley, H.E., Birgeneau, R.J., Reynolds, P.J., Nicoll, J.: J. Phys. C: Solid State Phys. 9, L553 (1976)

11. Stauffer, D.: Z. Physik B22, 161 (1975) 12. Reynolds, P.J., Klein, W., Stanley, H.E.: J. Phys. C: Solid State

Phys. 10, L167 (1977) 13. Stephen, M.J., Grest, G.S.: Phys. Rev. Lett. 38, 567 (1977) 14. Stephen, M.J.: Phys. Rev. B15, 5674 (1977) 15. Young, A.P., Stinchcombe, R.B.: J. Phys. C.: Solid State Phys. 8,

L535 (1975) 16. Leath, P.L.: Phys. Rev. B14, 5046 (1976) 17. Lubensky, T.C.: Phys. Rev. B15, 311 (1977), and in: Proceedings

of the Stat. Phys. 13 conference, Haifa 1977, to be published 18. Harris, A.B., Lubensky, T.C., Holcomb, W.K., Dasgupta, C.:

Phys. Rev. Lett. B35, 327 (1975) 19. Domb, C. in: Proceedings of the Stat. Phys. 13 conference, Haifa

1977, to be published 20. Stoll, E., Domb, C., Schneider, T., in: Proceedings of the Stat.

Phys. 13 conference, Haifa 1977, to be published 21. Harris, A.B., Kirkpatrick, S.: Phys. Rev. B16, 542 (1977) 22. Buyers, W.J.L., Pepper, D.E., Elliot, R.J.: J. Phys. C: Solid State

Phys. 6, 1933 (1973)

23. Holcomb, W.K.: J. Phys. C: Solid State Phys. 7, 4299 (1974), 9, 1771 (1976)

24. Krey, U., Schlichting, H.J.: Z. Physik B21, 157 (1975) 25. Kirkpatrick, S.: AIP Conf. Proc. 29, 141 (1976) 26. Thorpe, M.F., Alben, R.: J. Phys. C9, 2555 (1976); and AIP Conf.

Proc. 29, 250 (1976) 27. Holcomb, W., Harris, A.B.: AlP Conf. Proc. 24, 102 (1975) 28. Svennson, E.C., Buyers, W.J.L, Holden, T.M.: AIP Conf. Proc.

29, 248 (1976) 29. Krey, U., in: Proceedings of the Int. Conf. on Magn. Oxides and

Alloys, Haifa 1977. J. Magn. Magn. Mater. 7, 150 (1978) 30. Alben, R.: private communication 31. Alben, R., Kirkpatrick, S., Beeman, D.: Phys. Rev. B15, 346

(1977) 32. Funahashi, S., Moussa, F., Steiner, M.: Solid State Comm. 18,443

(1976) 33. Moussa, F., Villain, J.: J. Phys. C: Solid State Phys. 9, 4433 (1976) 34. Haydock, R., Heine, V., Kelly, M.J.: J. Phys. C: Solid State Phys.

8, 2591 (1975) 35. Rehr, J.J., Alben, R.: Phys. Rev. B16, 2400 (1977) 36. Stein, J.: Diplomarbeit, Fachbereich Physik der Universitgt

Regensburg, 1977, unpublished

V. Wagner Phys. Techn. Bundesanstalt Bundesallee 100 D-3300 Braunschweig Federal Republic of Germany

U. Krey Fachbereich Physik der Universigit Regensburg Universit/itsstr. 31 D-8400 Regensburg Federal Republic of Germany

Note Added in Proof. The structure in Figures 6b and 6c results mainly from localized ore "blocked" states; in particular, the peak near E/(2sJ)=2 corresponds to the blocked state depicted in Figure2 of Kirkpatrick, S., Eggarter, T.P.: Phys. Rev. B6, 3598 (1972).