HIGHLY RESOLVED X-RAY STRUCTURES AND THE ELECTRONIC STRUCTURES OF TANTALUM AND TUNGSTEN

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HIGHLY RESOLVED X-RAY STRUCTURES AND THE ELECTRONIC STRUCTURES OF TANTALUM

AND TUNGSTEN

G. Böhm, K. Ulmer

To cite this version:

G. Böhm, K. Ulmer. HIGHLY RESOLVED X-RAY STRUCTURES AND THE ELECTRONIC STRUCTURES OF TANTALUM AND TUNGSTEN. Journal de Physique Colloques, 1971, 32 (C4), pp.C4-241-C4-245. �10.1051/jphyscol:1971445�. �jpa-00214646�

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JOURNAL DE PHYSIQUE Colloque C4, supplLment au no 10, Tome 32, Octobre 1971, page C4-24 L

HIGHLY RESOLVED X-RAY STRUCTURES AND THE ELECTRONIC STRUCTURES

OF TANTALUM AND TUNGSTEN

G. BOHM and K. ULMER

Physikalisches Institut der Universitat (TH) D 7500 Karlsruhe, Engesserstr. 7

R6sum6. - La structure prks de la limite a haute frkquence du rayonnement ultra mou de freinage informe de la densite des Btats Blectroniques dans les domaines des bandes d'knergie inoccu- pBes en solides. On a mesurk les spectres du rayonnement de freinage pour les mktaux tantale et tungstkne a I'aide de cristaux courbks sphkriquement (dioctadecyladipate ; 2 d = 93,s tf) et dis- cut6 des structures fines, observks p r b de la limite a haute frkquence (longueur d'onde = 80 A

environ). Isochromates de la meme Bnergie ont 6tB mesurBes avec la meme disposition et sont comparBes avec ces spectres. Pour finir on compare les rksultats expkrimentaux aux densitBs des Btats thkoriques et trouve d'accord qualitatif.

Abstract. - Investigating the structure of t'he short wavelength limit of the continuous X-ray spectrum in the ultiasoft X-ray region provides information about the density of electron states in the unoccupied regions of the energy bands of solids. With synthetic self-grown and spherically bent dioctadecyladipate crystals (2 d = 93,s if) as dispersing devices the continuous X-ray spectra of Tantalum and Tungsten were measured. Fine structures near the short wavelength limit of about 80 ^i%are discussed. Measured X-ray isochromats of the same energy region are compared with the course of the theoretical densities of states above the Fermi edge and rigid band considerations for Tantalum and Tungsten, resulting in qualitative agreement.

1. Principle of experimental technique. - The clas- sical methods of X-ray spectroscopy are emission and absorption spectroscopy. The soft X-ray spectra obtained by these methods permit to draw conclusions on the course of the electronic density of states in the occupied and unoccupied regions of the energy bands of solids. But it must be stated, that soft X-ray spectra do not give a direct measure of the density of states [I].

In interpreting them due account must be taken of the relative transition probabilities, which involve the wave-function symmetries of the initial and final states. Further the considerable energy width of the participating inner levels in particular for atoms with higher atomic number [2], the Auger broadening of the conduction band levels [3] and many electron effects 141 are complicating the interpretation of emission and absorption spectra. But by the mentioned complicating factors also an advantage is brought about by offering greater possibilities in interpreting the experimental curves.

In the following two independent X-ray spectroscopic techniques shall be discussed, by which the course of the density of states in the unoccupied part of the conduction band can be determined. They are working without the participation of inner levels and may be regarded as a valuable completion and exten- sion of emission and absorption spectroscopy. These methods are

1) recording the fine structure near the short wave- lengt limit of the continuous X-ray spectrum and,

2) the isochromat spectroscopy 151.

Application of both methods will be described to the bcc transition metals Tantalum and Tungsten in the soft X-ray region in a wavelength range between 70 and 80 A. The experimental curves will be compared with theoretical densities of states for both metals above the Fermi edge and rigid band considerations. Figure 1 shows the principles of recording the continuous X-ray spectrum and

X-ray Spectrum lsochromot X-ray Spectrum lsochromol y = variable V= vor~oble hw = vorioble E'= vorrable V = const p = Cons t E' = const hw, = const

FIG. 1 . - Principle of the methods : Continuous X-ray spectrum and isochromat.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971445

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the isochromat in terms of one electron energy levels of the material to be investigated. The shaded sections indicate the occupied part of the conduction band with the Fermi edge EF. Of decisive importance for the practicability of our methods and related methods is the existence of characteristic energy losses of electrons in solids. For this point e. g. compare [5].

The continuous X-ray spectrum is generated by electrons of fixed energy E', which penetrate into the target and undergo radiative transitions to the unoccupied states above the Fermi edge E,. While the accelerating voltage V at the X-ray tube is kept constant, thelspectrum is surveyed successively by changing the Bragg angle q in such a way, that the target and the detector slit are moving symmetrically on the Rowland circle with respect to the crystal. The intensity of the radiative transitions ho is measured as a function of the pertaining Bragg angles. The short wavelength limit of the spectrum is given by the accelerating voltage at the X-ray tube. Yet this method suffers from some inherent pecularities, such as geo- metrical resetting of the apparatus during measurement, variation of the efficiency of the spectrometer with quantum energy and energy dependent self- absorption in the target, which can be avoided by a variant of this method.

This is the isochromat spectroscopy, the principle of which is shown in figure 1. In this case the initial state energy E' of the primary electrons is varied successively by changing the accelerating voltage. The adjoined spectrometer is operated as a monochromator, solely selecting a transition of fixed energy fio,, which is determined by the fixed Bragg angle and the lattice constant of the monochromator crystal. The intensity of this transition is measured as a function of the X-ray tube voltage. The transition with energy fro, will energetically be possible when E' - EF 2 ha,.

Two features of special importance for the subsequent interpretation and comparison of the experimental curves may be mentioned already here. In both cases, the measurement of the continuous X- ray spectrum and the isochromat, the primary electrons reach the same final states above the Fermi edge but starting from different initial states. Moreover the quantum energy tzw of the short wavelength limit and ^hooof the isochromat respectively represents a true parameter of the experiment, which can easily be varied by employing crystals with different lattice constants or different Bragg angles [6].

2. Apparatus. - The apparatus is a double focus- sing X-ray spectrometer, using a spherically bent crystal (*) as dispersing device [7] with a radius of curvature of 700 mm, which equals the diameter of

(*) As we recently learned Wassberg and Siegbahn [8] already successfully used spherically bent crystals for X-ray reflection in 1958.

the Rowland circle (Fig. 1). The material to be investigated is inserted as anode of the X-ray tube. The polycrystalline Ta and W amples of our experiment consisted of thin ribbons of a length of 10 mm and a width of 1 mm, mounted vertically on the plane of the Rowland circle at a symmetric position to the detector slit with respect to the crystal. The projection of the ribbon width in the direction of the analyzing crystal equals the slit width of the detector of 0.15 mm, thus utilizing the full breadth of the anode as source of radiation. During measurement the target temperature is 14000C with cleaning temperatures above 2 000OC. The cathode consisted of a Ta and a W wire respectively in order to avoid evaporation of impurity atoms on the anode. The cathode was heated intermittently and the radiation was recorded during the heating pauses by a Geiger-Miiller detector. Its gas volume is separated by a Formvar window from the oil-free vacuum of the spectrometer.

As dispersing device a spherically bent dioctadecyladipate crystal (OAO) of own growth [9] with a double interplanar spacing of 2 d = 93.8 A is used.

Single crystals of OAO of the order of 2 cm x 1.5 cm x 0.5 mm in size were grown from saturated solutions.

After placing a seed [(2 mm x 2 mm) in the saturated solution the temperature of the solution was lowered for 0.05 OCJday requiring a very close temperature control.

The crystals exhibit good mechanical properties, stability in vacuum and a pronounced cleavage.

Contrary to multilayer analyzers [lo], such as lead stearate, synthetic OAO crystals show a regular lattice.

They therefore seem very well suited for ultrasoft X-ray spectroscopy, as is indicated by the subsequent experimental results.

Finally the resolution power of the apparatus shall be mentioned briefly. A value of 7 000 was calculated after [7], During the experiment it is realized indeed because of the employment of a very narrow radiation source and detector slit. But it cannot be utilized to full extent on account of the Maxwellian energy distribution (x 0.4 eV) of the electrons emitted by the cathode.

3. Experimental results. - Figure 2 shows a compa-

FIG. 2. - Tantalum : The experimental curves are normalized to equal height of the maximum ; background subtracted.

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HIGHLY RESOLVED X-RAY STRUCTURES AND THE ELECTRONIC C4-243

rison of the continuous X-ray spectrum and the isochromat of Ta. Both curves have been normalized t o equal height of the maximum after subtracting the background. The given standard deviations include the background contribution. The short wavelength limit of the continuous X-ray spectrum corresponds to a quantum energy of E' - EF = 181.1 eV. This value contains a correction for the work function of the Ta cathode [ll]. A constant accelerating voltage of U = 177 V was applied to the X-ray tube for the measurement of the continuous X-ray spectrum. The quantum energy h a , = 177.3 eV of the isochromat has also been corrected for the work function of the Ta cathode [ll]. On the abscissa the values of the voltage applied are given. With respect to the statistical error both curves obviously exhibit the same appearance, a linear rise followed by a pronounced structure before the maximum.

Figure 3 shows a comparison of the continuous X- say spectrum and theisochromat ofW, again normalized

FIG. 3. - Tungsten : The experimental curves are normalized t o equal height of the maximum; background subtracted.

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t o equal height of the maximum after subtracting the background. The short wavelength limit of the continuous X-ray spectrum corresponds to a quantum energy of E' - EF = 171.5 eV. The quantum energy of the isochromat is ho, = 165 eV. Both values have been corrected for the work function of the W cathode [ l 11. Again the continuous X-ray spectrum and the isochromat look equal with respect to statistical error. Following a linear rise of intensity a struc- trure is observed before the maximum. Slight diffe- rences a t lower quantum energies and higher accelerating voltages respectively can be explained by apparative influences.

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4. Discussion of results. - From the comparison of the continuous X-ray spectrum and the isochromat of Ta and W respectively follows, that the intensity of the radiative transitions does obviously not depend on the initial energy level of the electrons. The conclusion is possible, that the transition probabilities involved can be regarded as independent of the initial

! X - roy Spectrum

I ⁱ ^{U =}^167V

i i - Ouonlum Energy I

170 165 160 Y

electron states for the employed polycrystalline sam- ples.

For considering this point in more detail, it is necessary to discuss, which physical quantity can be expec- ted to be reproduced by the experimental curves.

The calculation of the intensity is based on the following relation (in terms of one electron transitions) :

E' denotes the energy of the initial state and E the energy of the final state, correlated by the equation ho = E' - E. The factor (1 - f (E)) with the Fermi function f (E) indicates, that only transitions to unoccupied levels occur. P(kt, k) is the probability of the transitions involved. The integration has to be exten- ted over the surfaces S' of constant energy E' and S of constant energy E in k - space. By formally averaging over the energy surfaces E' = const and E = const in k - space, < P(kf, k) > ⁼P(Et, E), we can write (*) :

D(E) is the density of the final states. Assuming constant transition probabilities independent of E' and E the experimental curves should represent the density of the unoccupied states. We get finally :

In the last equation for the observed intensity I,,,(Ef, E ) we have made allowance for the instrumen- tal response by the indicated convolution with the apparative window A. The above assumption of constant transition probabilities is supported by the fact, that no inner levels with special atomic symmetries are participating in the emission process. In fact, there is also strong experimental evidence for this assumption. From the identity of the continuous X-ray spectra and the isochromats of figures 2 and 3 we can conclude, that the transition probabilities do not vary with the different initial states. Another example concerning the final states is given by an isochromat spectroscopic investigation of the 4 d transition metal alloy series Rh-Pd-Ag [12]. Accor- ding to this investigation the density of states of this alloy series can be very well interpreted in terms of a rigid band modell, which permits the conclusion, that the transition probability involved in the elementary process of the isochromat can also be regarded as independent from the final states of the electrons.

Thus one expects the isochromat and the continuous X-ray spectrum to picture the density of states above the Fermi edge, where the Fermi edge itself is broade- ned by the finite target temperature.

(*) For a more detailed discussion of this point see ref. [6].

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In figure 4 the measured Ta and W isochromats are

1 - - I compared with the coarse features of a theoretical

I ~ O X 15 density of states calculated for Ta by Mattheiss [13].

The Fermi edges of the isochromats are placed on the theoretical Fermi edges of Ta and W respectively,

'0 assuming a rigid band model1 for both metals. Satis-

0.5 fying qualitative agreement of theoretical and experi-

mental curves is found for Ta. The W isochromat seems

5 too broad, but the qualitative features of the theore-

E- E, tical curve are reproduced, too.

From these experimental results and their interpre-

I 0

-4 -2 o 2 tation it may be concluded, that high energy resolution experiments in the soft X-ray region can be done with crystals even on targets with high atomic number.

DIE)

(evatom~.l The continuous X-ray spectrum and isochromat

techniques seem to offer a more direct approach to the density of states above the Fermi edge than clas-

-10 sical absorption spectroscopy, resulting from the fact, that no inner levels are needed to participate in the transitions involved.

7 5 For our special examples we have brought forward

E- EM some arguments favouring constant transition proba-

e v bilities, necessary for the simple interpretation of our

I I I

- 4 - 2 o 2 - O experiments. Whether and how far this statement can

be generalized to other materials, in particular non FIG. 4. - Comparison of the experimental isochromats with transition metals, is an open question at the moment.

the theoretical densities of states. The origins of the energy We gratefully acknowledge support by the Deutsche scales are fixed at the isochromat maxima. Forschungsgemeinschaft.

References

[I] e. g. ROOKE (G. A.), J. Res. Nat. Bur. Stand. (U. S. A.), [6] BOHM (G.) and ULMER (K.), 2. Phys., 1969, 228,473.

1970,74A, 273. [7] EGGS (J.) and ULMER (K.), 2. angew. Phys., 1965, 20, [2] NELSON (G. C.), JOHN (W.) and SAUNDERS (B. G.), 118.

Phys. Rev., 1969, 187, 1. [8] WASBERG (G.) and SIEGBAHN (K.), Arkiv Fysik, 1958, [3] e. g. BLOCHIN (M. A.), in c( Rontgenspektren utzd 14, 1.

chemische Bindung )I, Leipzig, 1966. [9] BOHM (G.) and ULMER (K.), Z . angew. Phys. (to be [4] e. g. LONGE (P.) and GLICK (A. J.), Phys. Rev., 1969, published).

177, 526. [lo] HENKE (B. L.), Adv. X-ray Anal., 1966,9,430.

[5] ULMER (K.), Proc. Int. Symp. on X-ray Spectra and [Ill MERZ (H.), Phys. stat. sol. (a), 1970,1,707.

the Electronic Structure of the Substance, Kiev, [I21 EGGS (J.) and ULMER (K.), Z. Phys., 1968, 213, 293.

1968. [13] MATTHEW (L. F.), Phys. Rev., 1970, B 1,373.

DISCUSSION M.FABIAN. - YOU mentioned that the experimen-

tal results for the alloy series Rh-Pd-Ag agreed with a rigid-band model for the alloys, but you told us nothing about the nature of those experimental results ; was this simply a matter of the isochromat edge shifting, and of corellating the shift with electron-to-atom ratio ?

Answer. - Detailed arguments to this point are given in the following paper :

J. Eggs and K. Ulmer : 2. Phys. 1968, 213, 293.

There it is also shown, that absolute values for the density of states can be derived from these measurements.

M. KUNZ. -YOU ought to see the consequence

of energy loss of the electrons, which should be an important effect for 167 eV electrons.

a) Do you observe energy losses ?

b) Since the mean free path might be in the order of 10 A, did you make sure to have clean surfaces ? Answer. - Concerning your first question : We indeed observe Plasma losses in the isochromat and the continuous X-ray spectrum. For Ta and W the size of these discrete energy losses is of the order of 10 eV which leads to a repetition of the maximum of the curves at a voltage distance corresponding to the size of the energy loss. But now the fine structures in the curves are washed out on account of the considerable energy width of the plasma losses.

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HIGHLY RESOLVED X-RAY STRUCTURES AND THE ELECTRONIC C4-245 Concerning your second question : we used an

oil-free vacuum of mm Hg. The cleaning tempe- ratures of the targets were above 2 300 OK, the tempe- rature during measurement was 1 400 O K .

M. BAER. - Can you give a general expression for the transition probability involved in your isochro-

mat X-ray process ?

Answer. - Besides your own considerations published earlier there are no calculations known to us.

M. HAENSEL. - I think, the statement that the isochromats method is superior to absorption techni- ,ques should not be made too generally, if your energy resolution is 1 t o 2 eV. Absorption measurements can easily have energy resolutions of less than 0.1 eV as we have obtained in measuring the Pt 4 f edges (*).

In optical measurements however interchannel inter- actions indeed can often cause broadening of structures,

(*) Solid state Comm., 1969, 7, 1495.

so that I would prefer to define isochromats and absorption measurements to be complementary to each other.

M. ULMER. - 1) We consider it as a main advantage of the isochromat method to give a more direct picture of the electronic density of states in the conduction band (unoccupied states) than the other conside- red methods. This arises primarily from the fact that inner levels are not participating in the pertaining elementary process (the inner levels are often broade- ned-especially for materials with high atomic numbers) and in consequence atomic selection rules are of no importance.

2) For the question of the ((rigid band>> for the sequence Rh-Pd-Ag : Former measurements (J. Eggs and K. Ulmer : 2. Physik, 1968) have shown that

- within the experimental accuracy of this experiment

- a rigid band model can be ascribed very well to the sequence Rh-Pd and not so good to the sequence Pd-Ag.