PERIODIC OSCILLATIONS AND TURBULENCE OF HOT-CARRIER PLASMA AT 4.2 K IN n-GaAs

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PERIODIC OSCILLATIONS AND TURBULENCE OF HOT-CARRIER PLASMA AT 4.2 K IN n-GaAs

K. Aoki, T. Kobayashi, K. Yamamoto

To cite this version:

K. Aoki, T. Kobayashi, K. Yamamoto. PERIODIC OSCILLATIONS AND TURBULENCE OF HOT-

CARRIER PLASMA AT 4.2 K IN n-GaAs. Journal de Physique Colloques, 1981, 42 (C7), pp.C7-51-

C7-56. �10.1051/jphyscol:1981705�. �jpa-00221641�

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JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-51

P E R I O D I C O S C I L L A T I O N S A N D T U R B U L E N C E OF H O T - C A R R I E R P L A S M A A T 4.2 K IN n - G a A s

K. Aoki, T. Kobayashi and K. Yamamoto

Department of Electrical and Electronic Engineering, Faculty of Engineering, Kobe University, Rokkodai, Nada, Kobe, Japan

Résumé - Les oscillations périodiques et la turbulence du plasma de porteurs chauds causées par l'ionisation par choc de donneurs neutres et superficiels dans n-GaAs de 4.2 K ont été recherchées comme une fonction du champs électrique statique et de la densité de photo-excitation en utilisant la photo-excitation résonnante dans la longueur d'onde des lignes de (D ,X) et de (D ,X). Chaque rupture électrique

(le bruit spiciforme) contient les structures superposées de beaucoup d'oscillations périodiques et elle est expliquée par l'instabilité diffusive et induite du plasma de porteurs chauds fondée sur la transition de la première phase. Le mécanisme générateur du plasma de porteurs chauds est reconnu être un processus de Poisson, et la force fluctuante fortuite pour le déclenchement de l'instabilité est due au bruit des photons et au processus aléatoir de la formation et de la dissociation de l'exciton lié.

Abstract - Periodic oscillations and turbulence of hot-carrier plasma caused by the impact ionization of neutral shallow donors in n-GaAs at 4.2 K have been investigated as a function of static electric field and photoexcitation density, using resonant photoexcitation at the wavelength of (D ,X) and (D ,X) lines. Each electrical breakdown (spiky noise) contains superimposed structures of many periodic oscillations, and is phenomelogically explained by the diffusion-induced instability of hot- carrier plasma based upon the first-order phase transition. The generation mechanism of the hot-carrier plasma is found to be the Poisson process, and the random fluctuating force for the trigger of the instability is speculated to be due to the photon noise and the chance processes of bound-exciton formation and dissociation.

1. Introduction

Study of current-controlled negative resistance due to imapct ionization of 1 2^

neutral shallow donors in compensated materials of several semiconductors ' 'provides us a variety of informations about the energy relaxation mechanism, the impact ionization mechanism, and the carrier-plasma instability. Concerning the carrier- plasma instability, one may consider three types of the excitations; (1) homogeneous excitation of hot-carrier plasma, (2) the filamentary excitation of hot-carrier plasma and the mutual interactions between or among multi-filaments, and (3) the excitation which enforces mode patterns of the hot-carrier plasma in space and time.

The last type of the excitation has not yet been observed in semiconductor devices, but can be expected theoretically if we take into account the heat diffusion of hot electrons and the carrier diffusion in a couple of nonlinear differential equations which describe the heat balance and the impact ionization.

In this paper, we have investigated the carrier-plasma instability experimental- ly and observed quasi-periodic oscillations and turbulent state of hot-carrier plasma

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981705

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C7-52 JOURNAL DE PHYSIQUE

under weak photoexcitation i n high-purity n-GaAs a t 4.2 K. Our f i n a l goal i s t o c l a r i f y t h e type of the carrier-plasma i n s t a b i l i t y caused i n semiconductors, based upon the f irst-order phase t r a n s i t i o n . We also discuss about the microscopic origin of random fluctuating force which a c t s as a t r i g g e r f o r the i n s t a b i l i t y . 2. Experimental procedure

14 3

Samples used were epitaxial n-GaAs (n=2x10 /cm a t 300 K) grown on Cr-doped S.I. substrate. Thickness of t h e epitaxial layer was about 12 ym. Planar-type ohmic contacts were formed excellently by alloying Sn on the sample surfaces.

Typical dimensions of the sample surfaces were 5.5 mm i n length and 4 mm in width, respectively. All measurements were done a t 4.2 K. The l i g h t sources used were a 20 mW He-Ne l a s e r and a l s o a standard halogen lamp f o r the illumination of mono- chromatic l i g h t around band gap energy of GaAs. The current-voltage (J-E) charac- t e r i s t i c s were recorded on a recorder by applying d.c. voltage, with using a continuous motor drive. Speed of the motor drive was t y p i c a l l y ~ 3 0 mV/s. Power spectra of the photocurrent noise were observed a t a fixed e l e c t r i c f i e l d by using a spectrum analyzer YHP 8553 B.

3. Experimental r e s u l t s and discussions

The e l e c t r i c a l breakdown of the shallow neutral donors occurs a t around c r i t i c a l e l e c t r i c f i e l d of 4 ^Q6 V/cm under t h e dark and the weak photoexcitation. The c r i t i c a l e l e c t r i c f i e l d d i f f e r s s l i g h t l y from sample t o sample. Figure l ( a ) shows schematically the typical J-E curve observed under the dark and weak photoexcitation.

The onset of the e l e c t r i c a l breakdown ( i . e . , t h e lowest e l e c t r i c f i e l d a t which the spiky noise can be observed) s h i f t s t o lower e l e c t r i c f i e l d with increasing photoexcitation density J Each spiky noise has a typical waveform with the duration

P '

of about 10 ms as in Fig. 1 ( b ) , and shows the discontinuous jump and some hysteresis e f f e c t s . The spikes a r e randomly d i s t r i b u t e d in time, indicating t h a t the occur- rence i s probability-dependent a t a fixed e l e c t r i c f i e l d . The period between two events greatly depends upon the e l e c t r i c f i e l d and the photoexcitation density.

The power spectrum around 0 Hz showed a Lorentzian shape with an e f f e c t i v e relaxation time of about 2 ms 3)

.

The behavior i s analogous with the relaxation o s c i l - l a t i o n o r d i e l e c t r i c relaxation in semiconductors ( s o f t mode). Furthermore, the

Fig. 1 J n-GaAs 4.2 K

(a) The typica J-E curve observed under the dark and weak photoexcitation in n-GaAs at 4.2 K.

( b ) The typica waveforme of the spiky noise on the

oscilloscope, observed by vertical: 67 pA/div

a He-Ne laser exc'tation

t

with Jp=3.2 mW/cm

,

and horizontal: 5 ms/div

E=4.5 V/cm. E

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spike contains superimposed s t r u c t u r e s of quasi-periodic o s c i l l a t i o n s (hard mode).

Correspondingly, the power spectra show many d i s c r e t e l i n e s . Figure 2 shows the power spectra observed by a He-Ne l a s e r excitation with J =3.2 mw/cm2, as a function of e l e c t r i c f i e l d . Two d i s c r e t e lines a t around 1 MHz are due t o s t r a y P signals from radio frequencies. As i s shown i n t h i s f i g u r e , the d i s c r e t e l i n e spectrum changes t o continuous one above % 3.6 V/cm. The d i s c r e t e l i n e spectrum a t 3.1 V/cm indicates quasi-periodic s t a t e s , while the continuous spectrum suggests the

"turbulent s t a t e " of carrier-plasma i n s t a b i l i t y , by analogy of f l u i d system. A t present time, however, we include the classical Landau model i n the "turbulent s t a t e "

as well as chaotic phase 4

.

The turbulent s t a t e has been a l s o observed a t a fixed e l e c t r i c f i e l d , a s a function of photoexcitation density 3)

.

Figure 3 shows the power spectra observed by the resonant photoexcitation a t the (D',x) l i n e (818.5 nm) w i t h J

-

^{2 vW/cm}2

.

Number of the d i s c r e t e l i n e s

increases with increasing e l e c t r i c f i e l d . P- From the analysis, i t was found t h a t the d i s c r e t e l i n e s in each spectrum can be expressed by sum and difference of the higher order harmonics of two frequencies (incommensurate), namely by f = nfl + mf2. A t the e l e c t r i c f i e l d of 3.56 V/cm, two frequencies are f l = 3 5 kHz and f2=220 kHz, respectively. Above the e l e c t r i c f i e l d of 4 V/cm, a l l the frequencies were expressed by the combinations of some integers n=O ^%6, and m=O .L 6 . Change of the power spectra mentioned above greatly suggests t h a t both the s o f t - and hard-modes' excitations of c a r r i e r plasma occur a t the same time, and t h a t the continuous bifur- cation based on the f i r s t - o r d e r phase t r a n s i t i o n 4 ) occurs as a function of e l e c t r i c f i e l d and/or photoexcitation density. Physically, the carrier-plasma i n s t a b i l i t y can be explained by the current filamentation and by the mutual interactions between o r among multi-filaments. There i s , however, another p o s s i b i l i t y t h a t the c a r r i e r plasma forms mode patterns in space and time, which i s discussed l a t e r .

F i g . 2 Power s p e c t r a o b s e r v e d by a F i g . 3 Power s p e c t r a observed by

He-Ne l a s e r . t h e r e s o n a n t p h o t o e x c i t a t i o n

.

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G7-54 JOURNAL DE PHYSIQUE

The h o t - c a r r i e r plasma can be s e l e c t i v e l y generated a t t h e wavelength o f t h e (D',x) and (D+,x) l i n e s and a t t h e wavelength o f t h e e x c i t e d donor s t a t e s f o r t h e resonant f o r m a t i o n o f t h e bound e x c i t o n s 5 ) . Figure 4 shows s p e c t r a o f photocurrent n o i s e vs. e x c i t a t i o n wavelength, a t v a r i o u s p h o t o e x c i t a t i o n d e n s i t y . Scanning speed o f t h e monochromator was 10 A/min. The

arrows i n d i c a t e peak p o s i t i o n s o f emission

n-GaAs 4.2 K bands o f photoluminescence i n t h e same

sample. With v e r y weak p h o t o e x c i t a t i o n , no photocurrent n o i s e o r i n s t a b i l i t y can

be observed a t t h e wavelength o f f r e e -

+ +

1 ! I I

^{0.028 I}

e x c i t o n ground- and e x c i t e d s t a t e s , and above band gap energy, as i s shown i n t h i s

11 r ~ U l 11 dl

^{0.1 I}^.

f i g u r e . Furthermore, t h e number o f t h e s p i k y n o i s e per u n i t time a t a f i x e d

r l

^{0.2 I}^.

wavelength g r e a t l y depends on t h e photo-

e x c i t a t i o n d e n s i t y . 0.4 I~

Figure 5 shows a t y p i c a l long-time

exposure o f t h e c u r r e n t n o i s e on t h e In

-

recorder, a t t h e e l e c t r i c f i e l d o f 4.3 V

2

^LOO^PA

/cm and w i t h J z 60 nW/cm

.

Nine scans were c a r r i e d o u t t o t a l l y , each scan o f t h e P

819 818 817 816

monochromator being 5 minutes from 815.5

A (nm)

to 820.5 The observed 'pectrum Fig. 4 Spectra of the photocurrent noise

i s q u i t e s i m i l a r t o those observed w i t h on the recorder at 4.3 V/cm, as a function of excitation wavelength. The maximum photoexcitation density is I

-

^{2 pw/cm2.}

2'0

-

⁰^exp

.

.r(

E - theor.

. -

I 1

I I 1 1 I 0.1 0.01 ^h⁰ ^I ^,

1

-0.1

_C -

e,

-0.01

816 0.1 1 2

820 819 818 817 J ( p ~ / c m ~ )

A (nm)

I

I P l a I

Fig. 5 Long-time exposure of the photo- 0.1 1 10 20

current noise on the recorder, as a A

function of excitation wavelength. For

comparison, photoluminescence spectrum Fig. 6 Average density of the spiky observed by a He-Ne laser is also shown, noise per minute as a function of photo-

excitation density, at 4.3 V/cm.

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higher d e n s i t y of photoexcitation ( c f . Fig.4). From t h e s e r e s u l t s , we confirm t h a t t h e "bound-exciton catastrophe" a c t s a s a random f l u c t u a t i n g f o r c e f o r t h e impact i o n i z a t i o n of n e u t r a l shallow donors.

Figure 6 shows t h e average d e n s i t y of t h e spiky n o i s e per minute a t t h e wavelength of t h e (D',x) l i n e , a s a function of photoexcitation d e n s i t y . The minimum d e n s i t y was one pulse p e r t e n minutes with J

-

^{10 nW/cm}2

^.

The r e s u l t s p r e s e n t us a s t a t i s t i c a l problem concerning t h e microscopic o r i g i n of t h e random f l u c t u a t i n g P f o r c e . I f

we

consider some p a r t i c u l a r c e l l with a small volume 6V within which t h e c r i t i c a l f l u c t u a t i o n cause t h e i n s t a b i l i t y , t h e f l u c t u a t i o n of t h e bound-exciton d e n s i t y in t h e p a r t i c u l a r c e l l may be expressed by a simple Langevin equation.

Under t h e s t e a d y s t a t e c o n d i t i o n , t h e bound-exciton d e n s i t y 6N0 i n t h e c e l l i s given by 6VgT, where g i s generation r a t e a t s u r f a c e and T i s bound-exciton l i f e t i m e . The p r o b a b i l i t y t h a t t h e t o t a l number of t h e bound e x c i t o n s i s ~ N ( E ) during a r b i t r a - r y period of T % T + ^Ecan be expressed by t h e Poisson process because of t h e random f l u c t u a t i n g f o r c e c(T) ( <(T) = 6N(&)

-

6N0 ) . We a t t r i b u t e t h e random f l u c t u a t i n g f o r c e t o t h e photon noise of t h e l i g h t source and t o t h e chance process o r incom- p l e t e absorption f o r t h e bound-exciton formation. According t o t h e famous formula, t h e p o s s i b i l i t y i s expressed by,

where kT6N(&), A=6Vg=, and 6NC i s c r i t i c a l d e n s i t y of t h e bound e x c i t o n s which cause t h e i n s t a b i l i t y . Therefore, d e n s i t y of t h e spiky noise per u n i t time may be proportional t o t h e p r o b a b i l i t y given by e q . ( l ) . The s o l i d l i n e i n Fig.6 i s t h e c a l c u l a t e d r e s u l t of e q . ( l ) with using k>3, a s a f u n c t i o n of A. F i t t i n g between experiments and theory i s e x c e l l e n t .

F i n a l l y , we want d i s c u s s about t h e p o s s i b i l i t y of pure " t u r b u l e n t s t a t e " i n t h e carrier-plasma i n s t a b i l i t y which i s t h e most i n t e r e s t i n g problem i n t h i s paper.

For t h e p r e c i s e d e s c r i p t i o n o f t h e carrier-plasma i n s t a b i l i t y , i t i s necessary t o t a k e i n t o account t h e h e a t balance between long-wavelength a c o u s t i c phonons and hot e l e c t r o n s , a s well a s impact i o n i z a t i o n of neutral shallow donors. A couple of nonlinear d i f f e r e n t i a l equations can be obtained a s 3)

,

where two dimensional d i f f u s i o n s a r e assumed t o occur i n t h e d i r e c t i o n perpendicular t o t h e c u r r e n t flow. In e q . ( 2 ) , C i s t h e heat c a p a c i t y , K i s t h e thermal conduct- i v i t y , Te i s e l e c t r o n temperature, and F(Te,n,E) i s t h e nonlinear f u n c t i o n which d e s c r i b e s t h e energy balance between a c o u s t i c phonons and hot e l e c t r o n s under t h e e x t e r n a l e l e c t r i c f i e l d E and under t h e i l l u m i n a t i o n . In eq. ( 3 ) , n i s c a r r i e r

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C7-56 JOURNAL DE PHYSIQUE

d e n s i t y , D i s d i f f u s i o n constant o f m a j o r i t y c a r r i e r s , and G(Te,n) i s n o n l i n e a r term which describes impact i o n i z a t i o n , and thermal-recombination and - i o n i z a t i o n o f n e u t r a l shallow donors. The q u a n t i t i e s and E2 a r e random f l u c t u a t i n g f o r c e o f which importance has been a l r e a d y discussed.

Based upon t h e f i r s t - o r d e r phase t r a n s i t i o n , t h e d i f f u s i o n terms i n eqs. (2) and (3) w i l l p l a y i m p o r t a n t r o l e s f o r t h e carrier-plasma i n s t a b i l i t y ( d i f f u s i o n - i n d u c e d i n s t a b i l i t y ) . The necessary c o n d i t i o n f o r t h e soft-mode e x c i t a t i o n must be s a t i s - f i e d as f o l l o w s ,

where v i s sound v e l o c i t y , and <R> i s mean-free path o f t h e a c o u s t i c phonons, respec- t i v e l y . The c o n d i t i o n mentions t h a t t h e energy f l o w t o temperature b a t h acts as a long-range suppressive f o r c e ( i n h i b i t o r ) , w h i l e t h a t t h e impact i o n i z a t i o n a c t s as a short-range a c t i v a t i o n f o r c e ( a c t i v a t o r ) . A couple o f n o n l i n e a r equations ( 2 ) and ( 3 ) are s i m i l a r w i t h those i n t h e o s c i l l a t o r y chemical r e a c t i o n s and i n t h e b i o l o g i - c a l systems4), By t h e analogy w i t h o t h e r p h y s i c a l systems, t h e r e i s a p o s s i b i l i t y t h a t t h e c a r r i e r plasma forms mode p a t t e r n s i n space and time (e.g., c y l i n d r i c a l p a t t e r n s i n space and temporal q u a s i - p e r i o d i c o s c i l l a t i o n s ) , and i t cause the pure

" t u r b u l e n t s t a t e " as a r e s u l t o f continuous b i f u r c a t i o n s .

4. Conclusion

I n conclusion, we have i n v e s t i g a t e d t h e i n s t a b i l i t y o f h o t - c a r r i e r plasma i n n- GaAs a t 4.2 K. The main conclusions deduced were as f o l l o w s ;

1) P e r i o d i c o s c i l l a t i o n s and turbulence o f h o t - c a r r i e r plasma observed i n n-GaAs a r e e x p l a i n e d by t h e mutual i n t e r a c t i o n s between o r among m u l t i - f i l a m e n t s , o r by t h e f o r m a t i o n o f mode p a t t e r n s o f c a r r i e r plasma i n space and time, which i s deduced from d i f f u s i o n - i n d u c e d i n s t a b i l i t y .

2) D i s s o c i a t i o n process o f t h e bound e x c i t o n s acts as a random f l u c t u a t i n g force.

References

1) R.P. Khosla, J.R. F i s c h e r and B.C. Burkey ; Phys. Rev.

g,

2551 (1973) 2) W. Bludau and E. Wagner ; Phys. Rev.

E,

5410 ( 1 976)

3) K. Aoki, K. Miyamae, T. Kobayashi and K. Yamamoto ; J.J.App1 .Phys.

2,

L657 (1980), and see a l s o K. Aoki, K. Miyamae, T. Kobayashi and K. Yamamoto; Proc.

6 t h I n t . Conf. on Noise i n Physical Systems (1981, Washington) t o be published 4) H. Haken ; Synergetics, Springer-Verlag ( B e r l i n , 1977), and see a l s o

P.R. Fenstermacher, H.L. Swinney and J.P. Gollub ; J . F l u i d Mech.

94,

103 (1979) 5) K. Aoki, K. Miyame, T. Kobayashi and K. Yamamoto ; J . Phys. Soc. Japan

50,

357

(1981 )