Correlation of Orbach coefficient of spin-lattice relaxation with optical transition linewidths for trivalent erbium in lanthanum fluorid

Scott, P. L.

Scott, W. C.

Yen, W. M.

English

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We have measured the lifetime of the-first 
excited state of trivalent erbium in lanthanum 
fliioride, an find, for 0.05 percent erbium 
concentration, an inverse lifetime 
W = 7.0 k 0.9 X 1O1O sec'l. 
be 
value is observed to equal to the Grbach 
coefficient occurring in the spin-lattice 
For this salt, this 
4 
relaxation of the ground state doublet, recently 
measured by M. Schulz using electron s p i n  
resonance techniques. The equality of the 
inverse lifetime and the Orbach coefficient is 
in agreement with existing theory. 
We have also obsemed concentration dependent 
satellite structure in the ground state of 
trivalent erbium in lanthanum fluoride, which we 
tentatively attribute to exchange coupled pairs, 
We have sleo observed concentration dependent 
broadening in the optical transitions studied; 
this m y  be evidence of energy transfer between 
impurity Ions. 
-3- 
I* INTROIXiCTION 
It has been shown that lattice vibrations, or phonons, play a 
1 leading role in determining the widths of sharp optical transitions, 
and in spin-lattice (Tl) relaxation processes2 of rare earth 
impurities in solids. It has also been pointed out that correlations 
3 exist and may be made between ESR T1 studies and optical linewidth 
1 measurements. 
In particular, the process proposed by Orbach3j4 to account for 
the rapid exponential temperature dependence observed in 
relaxation times of certain rare earth impurities allows us such a 
direct correlation. Consider the energy level diagmm;&own in Fig. 1, 
drawn for the particular case of Er3+ 
T1 
in LaF3.5 The three levels 
shown in Fig. la are all Kramers doublets. 
transitions originating from the metastable ( 4S3/2) state may be 
observed in emission as shown. While both of the lines will be 
broadened by lattice strains, the 5402 8 line will have an additional 
broadening because of the short lifetimes of the states c and d. 
The two optical 
These lifetimes are determined by the probability for the spontaneous 
emission of a phonon of energy A 
the Er* ion to the ground state. 
is represented by W in Fig. la. 
accompanied by the transition of 
This latter nonradiative process 
Figure lb shows the lower two doublets when a magnetic field is 
applied, and share separately all the possible non-radiative , 
transitions involving a single phonon of energy CJ A In terms of 
. ' t' 
-. 
-4- 
these t rans i t ion  probabili t ies,  the lifetimes of the  states c 
and d may be writ ten as 'cC and -rd, where 
and 
6 For our samples, it may be sham t h a t  Wca = Wdb = W1 , and 
'cb Wda a w, 80 that 
~n general, W, f w,. 
Among the  many processes allowing electron spin- la t t ice  
relaxation between the two s ta tes  a and b, 
process3 is one involvirg the excited states c and d, i n  which 
phonon6 of approximate energy A are absorbed and em2tted. This 
spin l a t t i c e  relaxation process may be wri t ten 
the Orbach relaxation 
2 
- 
W l  + w2 
where TL is the spin- la t t ice  relaxation.time, and WL and W2 are 
as defined i n  the preceding paragraph. If Wl = W 2  , then the 
relaxation rate becomes Ti1 =I We - dkT , and the excited state 
(3)  
inverse lifetime is rol = W . In general, 'however, we note that 
.. -5- 
since 4 Wl W2/(Wl + W2) < (Wl + W2)# 
relaxation coefficient gives only an upper bound on the lifetime 
a measurement of the Orbach 
a 
of the excited state.  In the follawing paragraphs, we give 
experimental data showing that the non-radiative lifetime gotten 
by comparing optical line widths is essentially equal to the 
Orbach coefficient, thus implying that for our sample, W, = W 2  . 
. From the theoretical standpoint, we expect that W, = W 2  Each 
of these transition probabilities will be proportional to the 
square of a matrix element of the dynamic crystal field. 
of the large value of .J for the ground multiplet, and the l o w  
crystal symmetry, there may be as many as thirteen dynamic crystal 
field parameters, and the wave functions a, by c, and d will each 
Because 
. 
contain many different Jz states. 
probabilities will be a sum of twenty or thirty siniilar terms, and 
it is not likely that the two sums will be very different from each 
other. 
not likely to hold when the crystal symmetry is very law. 
Hence, each of the transition 
Put in another way, we say that rigorpus selection rules are 
We have also done some preliminary studies on the concentration 
dependence of some spectral lines of LaF3:Er3+ . We find a number 
of satellite levels accompanying the ground state of Er* which we 
attribute to ion pairs.7 
studied is also observed and may be indication that anergy transfer 
processes between impurity ions are occurring. 
Some concentration broadening of the lines 
a 
* 
.. -6- 
Unoriented single crystals of LaF, doped with various 
' concentrations (0.05s to 1%) of trivalent Erbium were gram 
by H. Muir of Varian Associates using a BriCCeman technique. 
Optical linewidths were studied both in absorption and 
. fluorescence using a high resolution 1.8 meter Jarrell-Ash 
EbertScanning Spectrometer. Detailed accounts of experimental 
procedures are given elsewhere and will not be repeated here. 1,9110 
The Er3+ impurity site in LaF, has C, symmetry; the 
optical spectrum of LaF3:Er3+ has also been published recently. 5 
It. is easily shown that in f1lis crystal configuration, all 
degeneracy, except the Kramers degeneracy due to time reversal 
symmetry, is lifted in the weak crystal field J states of the 
i o d 3  
and 
Er3+ in this coordination. Such measurements have recently been 
Er3+ 
All states of a given J-manifold are thus doubly degenerate 
ESR TI measurements may be made on the ground doublet of 
14 made by M. Schulz. 
To obtain correlation with the Orbach coefficient the optical 
transitions of interest are those shown in Fig. la. 
At 4.2'K, the fluorescence transition at 5288 2 reabsorbs 
heavily and produces considerable artificial weakening and 
broadening15 of the 5388 8 line even at low Er* concentrations. 
To circumvent this difficulty, it was necessary to study the widths 
of this line in absorption. The widths of the 5402 2 line were, 
.- 
. .  c 
-7- ... , 
on the other hand, studied in emission, since the ( 4115/.)c,d 
state, lying 51 cm'l above the ground state, is essentially 
unpopulated at ' 4.2OK. 
At sufficiently low temperatures, the linewidth of.optica1 
transitions show the effects of random crystal imperfections and 
any other, radiative or non-radiative, lifetime limiting 
mechanisms affecting the interacting states.  L 9  At 4.2OK, for 
the 0.05 percent LaF3:E+ sample, we find that 
= 0.24 f .02 cm'l A p3388 
* '5402 = 0.50 f -05 cm-l 
where the measured widths are the half-widths at half maximum. 
Now radiative lifetimes are normally too long to produce an 
observable optical. breadth; thus, in our case, at 4.2OK, the 
optical widths of the lines studied may be written 1 
and 
. -  
? 
.-. -8- 
where A 0 represent widths in energy units (cm”), and W is 
as defined by E q .  2. We now make two remarks: First, we assume 
that lattice strains contribute equally to both lines. While it 
may be possible to verify this assumption by observing the static 
shifts of these lines under the application of a uniaxial stress, 
such measurements have proved difficult, as the crystals tend to 
shatter rather easily. Second, we note that the second of EQs. 4 
is not strictly correct, as the strain broadening is expected to 
give a Gaussian line shape, while the lifetime broadening is 
expected to give a Lorentzian line shape. 
that the 
Our observations shaw 
5388 a line is Gaussian near the line center, but 
approaches a Lorentzian shape in the wings, while the 
follows a Lorentzian shape rather closely. As the best approximation, 
we assume that the 
numerical method to obtain the proper linewidth subtraction in order 
to obtain a value for W. ’’16 We find 
2402 2 line 
5388 8 line is purely Gaussian,. and use a 
W/2xc = 0.37 f .05 cm‘l, 
or 
w = T-J- = 7.0 * 0.9 x lolo sec-1 . 
(5) 
Using electron spin resonance techniques, Schulz ha8 measured 
the value of the Orbach coefficient 4 Wl W2/(W3. -i- W2) 
different crystals; he finds 
for four 
. 
. -9- 
The first and third samples listed in Eq. 6 were two of those 
on which our optical measurements were made. From the above data 
we conclude that, within the experimental error, for this sample, 
Orbach coefficient is equal to the inverse lifetime of the .excited 
state, thus implying that W, = W 2  . 
Optical measurements are made difficult at higher concentrations 
by concentration broadenhg and other effects which are dealtwith 
below. No correlation wae possible in the higher doped samples. 
Even at the lowest concentrations of Er3+ considerable 
satellite atructure appears accompanying the 5388 8 line 
(Figs. 2a and 3) .  No such structure accompanies the 5402 2 line, 
even though at higher concentrations the line becomes slightly 
asymmetric (Fig. 2b); thus, the structure must be connected with 
the ( 4 1 ~ ~ 1 2  >a,b ground s t a t e .  Since the relative intensity of 
the 
transition increases roughly as the square of the Er3+ concentration, 
we conclude that the structure is caused by exchange coupled pairs of 
Er3+ 
5388 8 satellites compared to the intensity of the central 
ions, much in the same sense as some of the satellite lines 
observed in Cr* in A& O3 and MgO. 15,18 
The structure of the 5388 1 line in beet seen in fluorescence 
(even though reabsorption broadening occurs) and is shown in Fig. 3 
-10- 
along with the splittings from the center transition. 
I 
We find some concentration broadening of the lines studied, 
this can be seen in Fig. 2. 
5402 8 line as a function of concentration are plotted in Fig. 4. 
The strain-uncorrected widths of the 
We believe that at higher concentrations, the broadening m a y  be 
caused by resonant energy transfer between adjacent impurity ions 
via a multipolar interaction. It is also possible that the a 
concentration broadening may be due to local lattice strains 
caused by the presence of nearby impurities. 
resonant, as wen as non-resonant,” energy transfer processes 
Detailed studies of 
could be carried out by considering optical linewidths, since 
, Dexter has s h m  that multipolar transfer interactions yield 
transition probabilities which lead to observable optical 
broadeningee a Such etudies are now being conducted in this 
laboratory. 
We wish to thank Mr. H. Muir of Varian Associates for making 
the crystals available to us. 
University of California, Berkeley, for allowing us the use of his 
data. We also would like to acknowledge numerous fruitful discussions 
with Bofessor A e  L. Schawlow and other members of his Stanford group. 
Acknowledgements are due M r .  M. Schulz, 
The high resolution spectrometer was made available by the 
Advanced Reeearch Projects Agency, through the Center for Materials 
Research at Stanford University. 
* -  . '  
.. 
. 
-11- 
h. M. Yen, W. C. Scott, and A. L. Schawlow, phys. Rev. (accepted 
fo r  publication) and references therein: 
i n  Quantum Electronics 111, edited by P. Grivet and N. Bloembergen 
(Columbia University Press, New York, 1964) p. 613. 
See also, A. M. Prokhorov 
*P. L. Scott and C. D. Jeffries, Phys. Rev. 32 (1962) and 
references therein.. 
'R. Orbach, Proc. Roy. Soc. (London) A264, 456 (1961). 
C. B. P. Finn, R. Orbach, and W. P. Wolf, Proc. Phys. SOC. (London) 4 
a 261 (1961). 
%. F. Krupke and J. B. Gruber, J. Chem. Phys. 
6 
1024 (1963). 
P. L. Scott, thesis, University of California, Berkeley, California 
1961 (unpublished) 
7S. Sugano and Y. Tanabe, i n  Magn etism, edited by G. T. Rado and 
H. Suhl (Academic Press, Inc., Hew York, 1963) Chap. 6. 
D. L. Dexter, J. Chem. Phys. 21, 836 (1953); D. L. Dexter and 
J. H. Schulman, J. Chem. Phys. 22, 1063 (1954). 
a 
'Go F. Imbusch, W. M. Yen, A. L. Schawlow, D* E. McCmber, and 
Me' D. Sturge, Phys. Rev. A1029 (1964). 
'OG. F. Imbusch, W. M. Yen, A. L. Schawlow, G. E. Devlin, and 
J. P. Remeika, Phys. Rev. (accepted for publication) 
lb. A. Jones, J. M. Baker, and D. F* D. Pope, Proc. Phys. SOC. 
(London) & 249 (1959). 
%. Y. Wong, 0. M. Stafsudd, and DI R. Johnston, Phys. Rev. 
990 (1963). 
0 .  
.. 
.. " .  . 
-12- 
13See, for example, C. J. Balhausen,'Introduction to Ligand 
' 
Field Theory, (McGraw-Hill Book Company, Inc., New York, 1962). 
14M. Schulh, to be published. We are greatly indebted to 
* Mr. Schulz for alluuing us the use of his data prior to their 
publication. 
. 15A. L. Schawlow, J. Appl. Phys. & 395 (1962). 
' %. W. Posener, Australian J. Phys. 12, 184 (1959). 
17L. F. Mollenauer and A. L. Schawlow, to be published. 
"G. F. Imbusch and A. L. Schawlow, to. be published. A detailed 
account of MgO:Cr* 
M. L. Report 1190, W. W. Hansen Laboratory of Physics, 
Stanford University, Stanford, California. 
spectra may be found in G. F. Imbusch, Thesis, 
"See, for example, L. F. Johnson, L. G. VanUitert, J. J. Rubin, 
and R .  A. Thomas, Phys. Rev. A493 (1964). ' 
'. . . 
. .  
I . .  
-13- 
F i g .  1 (a) Relevant optical transitions of LaF3:Er3+, sharing 
crystal field state positions. 
are Kramers doublets. 
(b) 
all possible transitions of the type W. 
AU. state8 represented 
The two lowest doublets in a magnetic field, sharing 
Fig. 2 Concentration dependence of the Er* (a) 5388 2 line in 
absorption and (b) 5402 8 line in emission. The 5388 2 
line shows concentration broadening and some satellite 
structure. 
by arrows in (b) . 
Asymmetries of the 5402 8 line are indicated 
Baselines are denoted by black lines. 
Fige 3 The 5388 8 line of Er- shown in emission along its 
satellites. 
to the central transition. 
Satellite positions are marked with respect 
Reabsorption broadening occurs 
in a l l  lines. 
Mg. 4 Uncorrected full widths of the 4.2OK 5402 2 transition 
plotted against nominal.concentration. Broadening appears 
to oocur at concentrations larger than 0.5% Er*. 
(4s3,2) - 
5402 
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Correlation of the orbach relaxation coefficient with optical linewidths- laf3-er3 plus

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