This paper presents a method for the computation of the magnetization and magnetic-field distributions of one transition in a perpendicular recording layer. It is used in a thick single-pole head keeper layer configuration to show that image-charge effects are significant and have a geometry-dependent impact on both height and gradient of the transition

Beusekamp, Martin F.

Fluitman, Jan H.

English

Beusekamp, M.F.

Fluitman, J.H.J.

University of Twente Research Information

1414 IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-21, NO. 5, SEPTEMBER 1985 SIGNIFICANCE OF IMAGE-CHARGE F I E W  ON THE WRITE PERFORMANCE OF A POLE-KEEPER HEAL). Martin F. Beusekamp and Jan H. Fluitman Abstract - This  paper  presents  a  method  for  the  computation of  the  magnetization  and  magnetic-field  distributions  of  one transition  in a perpendicular  recording  layer.  It is used  in a thick  single-pole  head  keeper  layer  configuration  to  show  that image-charge effects are significant and have a geometry- dependent  impact  on  both  height  and  gradient  of  the  transition. lNnWlXXIoN In the discussion concerning the merits of perpendicular recording  compared  to  longitudinal  recording,  Iwasaki  et  al. stated  that  the  interaction  between  the  magnetized  recording layer  and  the  main  pole  gives  rise  to  "steep  field distributions" [l], while on the  other  hand  Mallinson  et  al. stated that the field gradients of pole-keeper heads are "virtually identical" to those of ring heads [2]. In this paper,  the  exact  shape  of  a  magnetization  transition  written  by a pole-keeper head of given geometry is computed and the accompagnying  fields  are  analysed. The  geometry of the  investigated  recording  system is given  in fig. 1, which  shows  the  right-hand  edge  of  a  thick  single  pole head (e.g. Ohtsubo [31) opposite a perpendicular recording layer  with  thickness  t  and  separated  from  this  main  pole  by  an air  gap  g.  The  recording  layer  is  backed  by  a  keeper  layer  with an  assumed p=oD and  has  an  initial  magnetization M, in  the direction as drawn.  The  recording  layer  material  is  assumed  to have  a rectangular  (intrinsic)  hysteresis  loop  with &>& (fig. 2). The head can induce a field opposite to &, thereby writing one transition in the region near the edge  of  the  head. RECORDING PROCESS IN THE DEEP GAP REGION In  the  region  inside  the  head  and  far  from  the  edge  (fig. l), Maxwell's  laws  can be written  as: g.% + t.H = nI (1) H g = H + M  (2) From  these  equations  it s clear  that,  without  a  magnetomotive force driving the head (nI=O), % will  not  equal  zero  and will be opposite  to  the  self-demagnetizing  field H. Eliminating % from (1) and  (2)  will  give: Manuscript  received  February  27,  1985. The  authors  are  with mente University  of  Technology, P.O. Box  217, 7500 AE Enschede,  The  Netherlands. 'Ihe  first  term  in (3)  is the  head  field  as  it  would  occur  in the  absence  of  a  recording  layer [ 2 ] ,  the  second  term  is  the self-demagnetizing  field  in  the  layer (H = -M), while  the  third term  must be ascribed  to  the  image-charge  field,  caused  by  the reflection  of  magnetic  charge  of  the  recording  layer  in  the ple piece  and  keeper  layer.  From ( 3 ) ,  the  significance  of  the image-charge  field  is  obvious,  especially  when  g/t  is small. In fact, it causes the high efficiency of the perpendicular recording  head  in  the  presence  of  a  keeper  layer. Equation (3 )  can  be  rewritten  as: which  gives us, apart  from  the  hysteresis  loop  of  fig. 2, a second  relation  between M and H in  the  recording  layer.  It is clear  that  the  presence  of  the  head (g<m) causes  a  steepening of  the  slope  of  the  initial  relation M = -H, while  a  magneto- notive  force  nI  driving  the  head  causes  a  vertical  shift  of  the line  (fig. 2). The  initial  situation  being A, the  positioning of  the  head  will  decrease  the  magnetic  field  in  the  recording layer until B is reached. A magnetomotive force nI=t'.Hc will  restore  situation A, a  further  increase  of  nI  causes  a change  of  magnetization,  for  instance  until C is reached. A decrease  of  nI  to  zero  will  then  result  in  situation D and removing  the  head  will  cause  the  magnetization  to  relax  to E. Fig. 2. Assumed intrinsic  hysteresis  loop  of  the  recording layer. RECORDING PROCESS NEAR THE  EDSE OF THE HEAD General. Near  the  edge  of  the  head,  the  image-charge  field is still significant.  Its  computation  is  not  trivial,  and  has so far been  neglected  in  literature.  The  image-charge  influence  can  be derived  from  a  computation  of  the  magnetization  distribution  in the edge region, which is based on the assumption that H=% in every part of the recording layer during writing. This is shown by Wielinga et al. [41 and was already illustrated  by  fig. 2. The recording layer region near the edge of the head is subdivided  into  n  narrow  stripes  having  magnetic  fields Hi 0018-9464/85/0900-1414$01.0001985 IEEE 1415 and  magnetizations Mj (l<=j<=n). We consider vectors €I and M respectlvely.  The  magnetic  field  in  each  stripe mnsists of three contributions: 1) the local head field Hh, 2) the  contribution  Hf of the  recording  layer regions  outside  the  subdivided  area,  where  the  magnetization is assumed  to be constant,  and 3) the  contribution I$,,, caused by the  magnetization  in  all n considered  stripes.  The  vector I+,,, consisting of the n values of. Hm, can therefore be written  as [DI.n, where. Dl is an  n*n-matrix  containing  the mgnetic interactions  between  each  pair  of  stripes. The sum of the  field  contributions,  which  as  stated  must  equal €I,-, can now be written  as: H1r-**tHn and  MII. .. ,Mn  as  twon-dimensional Bh+FIf+ Dl.U=& (5) where H, is a vector  containing n elements  with a numerical value of &. This implicit solution of M can be made explicit  by  inverting [Dl and  rearranging  the  terms: H = Dl-1 {H, - lk, - (6) Now, M can easily be calculated if [Dl, lk, and af are known. Without  imige  charges  taken  into  account. If we neglect the image charges, the only influence the magnetization  Mi  n  the  i-tb  stripe  performs  on  the magnetic  field H. in  the  j-th  stripe' is the  self- demagnetization,  wdich  can  be  calculated  by  using Coulomb's law for  magnetic  charges: c= p . F m  4 ar3  (7) A unit  surface  charge  Mdxdz  on  the  top  surface  and a unit charge  -Mdxdz  on  the  bottom  surface  of  the  recording  layer together induce a field dH at a specified point (x,y,z), which can be calculated w i d  (7) (fig. 3 ) .  Note that the concentration  of  magnetic  charge  in  the  top  and  bottom  surfaces of  the  recording  layer  implies  the  assumption  of a homogeneous mgnetization distribution within each of the stripes. Integrating  Mdxdz  over  the  top  and  bottom  surface  of  the  i-th stripe (xi < x < Xi+Ax and -a< z <a) and integrating dHy  over  the  face of the j-th  stripe  (xj < x < Xj+AX and 0 < y < t) gives us the possibility to compute the contribution of the magnetization in the i-th stripe  to  the  magnetic  field  in  the  j-th  stripe.  Evaluation  for all  combinations  of i and j will  yield [Dl. It is clear  that dij  only  depends  on  the  distance  between  the  i-th  and  j-th stripe, so ID] will be a symetric tensor. af can be calculated  similarly  by  considering  the  recording.  layer  regions outside the subdivided area as "stripes" with an infinite width. The top surface of the keeper layer is an equipotential surface,  just  as  the  symnetry-axis  of a ring  head is (fig. 4 ) .  The  given  geometry  can  therefore be considered  as  one  half  of aring  head,  with  the  recording  layer  running  through,  instead  of Y X i  X i + A X  i-th stripe ~ X .  X . + A X  j-th  stupe Fig. 3. Determination  of  dij  without  taking  the  image charges  into  account. f along, the gap. Therefore, l+, can be approximated by a known analytical  function  (Karlqvist [5], Szczech [ 6 ] ) ,  but  can also  be  calculated  as  discussed  in  the  next  subsection. With  image  charges  taken  into  account. For  the  compbtation  of Dl including  the  influence  of  image charges, the contours of the pole pieces of fig. 48 are transformed  into a straight  line  by a conformal  mapping  method (fig. 4) .  The  top  surface  of  the  recording  layer  will  map  into a curve  in  the  right  half  plane  and  any  magnetization  on  that surface  will be mirrored  by  the  above  mentioned  straight  line. Every  dij  can be recomputed  by  transforming  the  nd surfaces  of  both  the  i-th  and  j-th  stripes,  assuming  unity magnetic  charges  with  opposite  signs  in the e d  surfaces  of the i-th  stripe  and  deriving  the  potential  difference  between  the end  surfaces  of  the  j-th  stripe  in  the  transformed  domain.  This ptential difference  will  not be affected  by  the  (inverse) transformation, so that dividing it by the recording layer thickness t will result in dij. Now, dij will depend on tath  the  distance  between  the  i-th  and  j-th  stripe, as well  as their positions with respect to the edge of the head. Therefore, the syrrmetry of D] will be disturbed. Bf can again be computed,as the effect  of  infinitely  sized  "stripes" and I+, will easily result from a computed potential difference  in  the  transformed  domain  with a known potential difference force on  the  transformed  parts  of the pole  pieces of  the  equivalent  ring  head. rnl t ' Fig. 4. A) Given  pole-keeper  head  geometry. B) Equivalent  ring  head  geometry. C) Resulting  geometry  after  conformal  mapping. RESULTS AND DISCUSSION Figure 5 illustrates the significance of the image-charge field.  Without  it,  the  magnetization  tends  to  peak  near  the transition,  due  to  the  fact  that  the  recording  layer  near thetransition is magnetized  rather  than  demagnetized  by  its  right- hand  part.  The  image-charge  field  partially  compensates  the self-demagnetizing field. This can be considered as an enhancement  of  the  effective  field  underneath  the pole as well as a reduction of the self-demagnetization within the transition.  The  r sult is the  disappearance  of the magnetization  peak  (unless  g/t<<l)  and  an  increase  of  the deep gap  magnetization.  It is clear  that  the  smaller  g/t is chosen, the more influence  the  image-charge  field  will  have.  It  was observed that with g/t>>l (an unpractical situation), the magnetization  distributions  computed  with  and  without  the  image charges,  although  computed  by  essentially  different  algorithms, were  exactly  the same. Figures 5 and 6 illustrate  the  effect for  two  g/t-ratios. After  computing a transition  with  image  charges  taken  into account,  the  self-demagnetizing  field  that is caused  by  the mmputed magnetization distribution can be calculated with Coulomb's law  as  mentioned  previously.  The  contribution  of  the image-charge  field  can  now be calculated  by  subtracting  both this  self-demagnetizing  field  as  well as the  head  field  from Q. The  result  of  this  split is shown  in  fig. 7. Adding  the image-charge field to the head field results in a higher effective  head  field  and  head-field  gradient,  causing a higher and steeper transition, as already shown (fig. 5 ) .  The gradient  of  the  image-charge  field  itself  however, is smaller 1416 than  that  of  the  head  field.  Normalization  on  the deepgap mgnetization therefore, results in a smaller gradient of mgnetization  (fig. 8). -7 + + + + * + + + + * + + + +  A t = 0.7 pm g = 0.3 prn - u _I N -  z -  .. o_ 2 - -  ............. E 1 1  A - W I T H  1 IMAGE-CHARGE E -  .* E-WITHOUT  F IELD N l , , , , , l , ~ ~ l ~ ~ ~ ~ ~ l ~ ~ ~ ~ ~ l  ' - 3 - 2  -I 0 1 2 3 4 5 6 7 8 POSITION Ipm.O=edge of head1 Fig. 5. Magnetization  distribution  with  and  without  the image-charge  field  taken  into  account  (t=0.7pm, g=O.3pm). The  deep-gap  head  field  equals 2 Hcin  both  cases. t =0.3 pm g =0.7 p m  g N a -#... .......... A-W  ITH  IMAGE-CHARGE + A z - + * * + + + + + + + + + +  * -  B .+ + w 0 -  0 z I T -  a N , , , , , , , I  I I I I I I I  I I I I 1  ' - 3 - 2  -1 0 1 2  3 4 5 6 7 8 POSITION Ipm,O=edge of headl Fig.  6.  Magnetization  distribution  with  and  without  the image-charge  field,taken  into  account  (t=0.3pm, w.7 pm) . The deepgap head  field  equals  2  Hc in  both  cases. A-HEAD FIELD, E-SELF-DEMAGNETIZING FIELD C-IMAGE-CHARGE  FIELD - 3 - 2 - 1  0 1 2  3 4 5 6 7 8 POSITION lprn, O=edge of headl Fig. 7. Contribution  of  the  head  field,  the  self-demagneti- zing  field  and  the  image-charge  field at a  deep-gap head  field  of 1.6 Hc (t=0.7 pm, 94.3 pm). m C  "1 t =0.7pm g =0.3 pm Fig (v - 3 - 2 - 1  0 1 2  3 4 5 6 7 8 , , , , , ( , , , , , , , , , , , , ~ , , 1  I POSITION lpm, O=edge of headl . 8. Magnetization  distribution  with  and  without  the image-charge  field  taken  into  account  (t=0.7pm, g=0.3pm).  The  deep-gap  head  fields  are  such,  that the  deep-gap  magnetization  equals 2 Hc  in  both cases. The  results  are  given  with  the  activated  head  in  position.  The removal  of  the  head  will  cause  the  total  field  to  decrease. Under  the  given  assumption,  a  rectangular  intrinsic  hysteresis loop  with  rectangular  minor  loops,  all  magnetization  values will remain  constant  under  the  decreasing  field  until  the  left- hand  edge  of  the  hysteresis  loop is reached.  The  relaxation process  which  will  then  take  place,  can  also be described  by  an algorithm.  It is clear  that  the  magnetization  of  the  part  of the  recording  layer  which  is  in  the  deep-gap  region  during writing,  will  ultimately  relax  to  M=Hc.  This  will  cause  the typical  self-magnetization  peak  in  the  transition  region  to occur, as was  already  observed  in  a  preliminary  run  of  the relaxation  algorithm.  Results  will be published  in  the  near future. CCNCLUSIDNS It can be concluded from the complted magnetization and mgnetic-field  distributions  of  a  transition  in  a  perpendicular recording layer that the effects of image charges are significant and geometry-dependent. Their influence can be considered as an  enhancement  of  the  magnetomotive  force  by which the head is driven, resulting in higher and steeper transitions.  On  the  other  hand,  the  effect on the height  of  the transition  is larger  than  that  on he  gradient  of mgnetization.  Therefore,  if  the  magnetization  under  the  head is normalized,  a lower maximum  gradient  in  the  transition is found.  The  self-magnetization  peak,  a  typical  feature  of a perpendicularly recorded transition, is not present during writing,  but  will  appear  after  the  head  is  removed  from  the recording  layer. S. Iwasaki,  Y.  Nakamura, S. Yammto, K. Yanakawa.  "Perpen- dicular  Recording  by a Narrow  Track  Sinqle  Pole  Head". 1.E.E.E.  Trans. on  Magn., Vol. MAG-19, NO. 5, PP. 1713-5, September  1983. J.C. Mallinson, H.N.  Bertram. "On the  Characteristics  of Pole-Keeper  Head  Fields".  I.E.E.E.  Trans. on Magn., Vol. M4G-20, No. 5, pp.  721-3,  September  1984. A. Ohtsubo,  Y.  Satoh.  "Thick  Non-Parallel  Single  Pole  Head for  Perpendicular  Recording  from  One  Side".  I.E.E.E.  Trans. on  Magn., Vol. MAG-18,  No. 6, pp.  1173-5,  November  1982. T. Wielinga, J.H.J. Fluitman,  J.C.  Lodder.  "Perpendicular Stand-Still  Recording  in  CoCr-Films".  I.E.E.E.  Trans.  on Magn., Vol. MAG-19,  No. 2, pp. 94-104,  March  1983. 0. Karlqvist.  "Calculation  of  the  Magnetic  Field  in  the Ferromagnetic  Layer  of  a  Magnetic  Drum".  Trans.  Royal  Inst. of  Techn.  Stockholm, Vol. 86,  pp.  1-27,  1954. T.J. Szczech, D.M. Perry, K.E.  Palmqvist.  "Improved  Field Equations  for  Ring  Heads".  I.E.E.E.  -Trans.  on  Magn., Vol. M4G-19, No. 5, pp.  1740-4,  September  1983. 

Significance of image-charge fields on the write performance of a pole-keeper head

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Significance of image-charge fields on the write performance of a pole-keeper head

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