Physica status solidi / A.: Volume 51, Number 1 January 16 [Reprint 2021 ed.] 9783112495209, 9783112495193

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plxysica status solidi (a)

z w «

« < « > W S.

© «-
700 K . Petrov suggested a model connecting the lattice structure of (3-boron and its thermal conductivity [80]. There are two parts of the phonon spectrum in fi-boron: A > a0 and A , C3-137 (1975).

[6] B. TELL, J. L. SHAY, and H. M. KASPEE, Phys. Rev. B 4, 2463 (1971). [ 7 ] G . C. BHAB a n d R . C. SMITH, p h y s . s t a t . sol. (a) 1 3 , 1 5 7 ( 1 9 7 2 ) . [ 8 ] J . L . REGOLINI, S . LEWONCZUK, J . RINGEISSEN, S . NIKITINE, a n d

C. SCHWAB, p h y s .

sol. (b) 55, 193 (1973). [9] B. TELL and H. M. KASPEE, Phys. Rev. B 7, 740 (1973). [10] N. YAMAMOTO and T. MIYAUCHI, Bull. Univ. Osaka Prefect., Ser. A 23, 147 (1974).

stat.

[ 1 1 ] B . T E L L a n d P . M. BBIDENBAUGH, P h y s . R e v . B 1 2 , 3 3 3 0 ( 1 9 7 5 ) . [ 1 2 ] P . ROCHON, E . FOETIN, J . P . ZIELINGER, a n d C. SCHWAB, J . P h y s i q u e , Suppl. 8 6 , C 3 - 6 7 ( 1 9 7 5 ) .

[13] B. SEBMAGE and M. Voos, Phys. Rev. B 15, 3935 (1977). [14] S. SUGAI, J. Phys. Soc. Japan 43, 592 (1977).

62

W.

HORIG

et al.: Temperature Dependence of t h e Absorption Edge in CuGaS 2

[15] J . P . ZIELINGER, C. NOGUET, a n d M. TAPIERO, I n s t . Phys. Conf. Ser. 35, 145 (1977). [ 1 6 ] W . H O R I G , H . N E U M A N N , H . SOBOTTA, B . S C H U M A N N , and G . K U H N , Thin Solid Films

48,

67 (1978). [ 1 7 ] G . D . B O Y D , H . M . K A S P E R , a n d J . H . M C F E E , I E E E J . Q u a n t u m Electronics 7 , 5 6 3 ( 1 9 7 1 ) . [18] E. J . JOHNSON, Semicond. Semimetals 3, 153 (1967). [ 1 9 ] 1 . 1 . B O I K O , Fiz. tverd. Tela 3 , 1 9 5 0 ( 1 9 6 1 ) . [20] D. A U V E R G N E a n d J . CAMASSEL, phys. stat. sol. (b) 44, 687 (1971). [ 2 1 ] R . G . G L I N S K I , K . S . SONG, and J . C. W O O L L E Y , phys. stat. sol. (b) 4 8 , 8 1 5 ( 1 9 7 1 ) . [ 2 2 ] P . B . A L L E N a n d V . H E I N E , J . Phys. C 9 , 2 3 0 5 ( 1 9 7 6 ) . [23] J . CAMASSEL a n d D. A U V E R G N E , Phys. Rev. B 1 2 , 3258 (1975). [24] R. G. H U M P H R E Y S a n d B. R. P A M P L I N , J . Physique, Suppl. 36, C3-159 (1975). [ 2 5 ] A . R A U D O N I S , V . S . GRIGOREVA, V . D. P R O C H U K H A N , and A . S I L E I K A , phys. stat. sol. (b) 5 7 , 4 1 5 (1973). [ 2 6 ] N . A . GORYUNOVA [27]

Moscow 1974. J. L . SHAY, E .

and Yu.

BUEHLER,

[ 2 8 ] J . KAVALIAUSKAS,

A . VALOVA,

and J .

A . SILEIKA,

H . WERNICK,

Phys. Rev.

N . A . GORYUNOVA,

Letters A 33, 49 (1970). [29] I . P. A K I M E N K O , A . S. B O R S H C H E V S K I I , a n d V. S. [ 3 0 ] E . L . KARASEVA, [31]

Semiconductors, Izd. Sovetskoe

A2B4CI

G . A . SIKHARULIDZE,

B 4, 2479 (1971).

E . I . LEONOV,

IVANOV,

Radio,

and

V . M . ORLOV,

Phys.

Fiz. Tekh. Poluprov. 7, 144 (1973).

V . M . TUCHKEVICH,

Y U . I . UKHANOV,

and

Y u . M.

SHMARTSEV, Fiz. Tekh. Poluprov. 1, 276 (1967). B . M O N E M A R , Solid State Commun. 8 , 1 2 9 5 ( 1 9 7 0 ) .

[32] Y . F . TSAY, B . GONG, S. S. MITRA, a n d J . F . VETELINO, P h y s . R e v . B 6, 2 3 3 0 (1972).

a n d W . J U N G E , phys. stat. sol. (a) 3 4 , K 3 9 ( 1 9 7 6 ) . [34] J . L. SHAY and J . H . WERNICK, Ternary Chalcopyrite Semiconductors — Growth, Electronic Properties, and Applications, Pergamon Press, 1975. [35] D. W E A I R E and J . N O O L A N D I , J . Physique, Suppl. 36, C3-27 (1975). [36] V. H E I N E and J . A. VAN V E C H T E N , Phys. Rev. B 13, 1622 (1976). [37] V. CAPEK, phys. stat. sol. (b) 81, 571 (1977). [38] A. GOLDMANN, phys. stat. sol. (b) 81, 9 (1977).

[33] H . NEUMANN

(Received

May

22,

1978)

A. YU. LEIDERMAN and P. M. KARAGEOKGY-ALKALAEV: Theory of Diode Structures

63

phys. stat. sol. (a) 51, 63 (1979) Subject classification: 14.3.3; 14.3.4; 22.2.1 Institute of Technical Physics, Academy of Sciences of the Uzbekian SSR,

Tashkent

On the Theory of Current-Voltage Characteristics of Semiconductor Diode Structures with Strong Carrier Accumulation By A . Y u . LEIDERMAN a n d P. M . KARAGEORGY-ALKALAEV The theoretical treatment of dc forward-biased n + - n - n + and p + - n - n + semiconductor diode structures is obtained for the case of strong carrier accumulation resulting in a field-opposed diffusion in the n-base region. The ambipolar drift is limited by the injection modulation of population of deep compensating traps N t and the well defined distinctive current saturation (sublinear region on I-V curve) of the type V(I) ~ V0 exp [A(0, AT) I] occurs. Just in this saturation region the topic voltage responsivity to temperature variations AT and incident photoexcitation 0 must be reached. The agreement of the developed theory with GaAs n ' - n - n 1 structure properties is discussed. IIPOBENEHO TeopeTHHecKoe HCCJieaoBaHHe CTa-nmecKHX TOKOB B n + - n - n + - H p + - n - n + HHORHMX cTpyKTypax, CMemeHHbix B nponycKHOM HanpaBJieHHH, B ycjioBHHx n p e o ô j i a NAHHH B HHX 3(J>(j>eKTa aKKyMyjiHKHH, oôycjioBJiHBaiomeii BCTpeiHbie HanpaBJieHHH 6HnojiHpHtix HH$$y3HH H npeii(|>a. B cjiynae Korjia ÔHnojiHpHbiii npeii(J> onpenejiHeTCH HHmeKHHOHHOH MOHyjiHiiHeft 3anoJineHHH rjiySoKHX KOMneHCHpyromnx jiOBymeK Nt Ha B A X B03HHKaeT oôJiacTb cyôJiHHeiiHoii 3aBHCHM0CTH TOKa ( „ H a c b i m e H H e " Tona) Tuna V(I) ~ V0 exp \_A{&, AT) /]. PÏMeHHO B 3TOii cyôJiHHettHOH oôJiacTH B A X jiocTHraeTcn HaHÔOJlbUiaH

BOJlbTOBaH

HyBCTBHTejlbHOCTb

HCCJieHyeMblX

CTpyKTyp

K

H3MeHeHHHM

TeMnepaTypbi AT H $OTOB036yHiHeHHio 0. OôcywmaeTCH corjiacne pa3BHToii TeopHH c x a paKTepncTHKaMH GaAs n + - n - n + - c T p y K T y p . 1. I n t r o d u c t i o n T h e m a t t e r of this p a p e r is t h e t h e o r e t i c a l t r e a t m e n t of t h e dc c u r r e n t - v o l t a g e characteristics of t h e d i o d e structures w i t h s t r o n g carrier a c c u m u l a t i o n . Such a s i t u a t i o n m a y b e t h e case e.g. in t h e high-resistive base r e g i o n of f o r w a r d - b i a s e d p + - n - n + a n d n + - n - n + structures ( F i g . 1) w h e r e d u e t o s t r o n g carrier a c c u m u l a t i o n t h e i n j e c t e d plasma d e n s i t y increases g o i n g f r o m a n o d e t o w a r d c a t h o d e (i.e. p(w) > p ( 0 ) a p p e a r s ) g i v i n g rise t o f i e l d - o p p o s e d d i f f u s i o n of f r e e carriers. 2. Basic E q u a t i o n and B o u n d a r y Conditions T h e basic e q u a t i o n d e f i n i n g t h e b e h a v i o u r of t h e i n j e c t e d p l a s m a in h i g h - r e s i s t i v e n-base r e g i o n of p + - n - n + a n d n + - n - n + d i o d e structures is t h e a m b i p o l a r c o n t i n u i t y equation (1) Here 1

Dipolar d i f f u s i v i t y

64

A . Y u . L E I D E R M A N a n d P . M . KARAGEORG Y - A L K A L A E V

Fig. 1. Schematic drawing of the diode structure with strong carrier accumulation, a) p + - n - n + structure and b) n + - n - n + structure

o

is the ambipolar drift velocity, Uv — (p — pn )l rP — 0(0) is the total intensity of generation-recombination processes, generally containing besides the ordinary recombination (described with the usual lifetime constant r p ), the carrier photogeneration term 0(0). In this nonlinear second-order differential equation each term has a simple physical interpretation connected with a certain plasma relaxation regime. For the semiconductor diode structures with monotonical variation of carrier concentration, in which as a consequence of injecting contact properties the fieldopposed diffusion appears, a distinct situation takes place when the influence of G - R processes on the plasma relaxation is negligible and thus n

d2

| l

i

!

d a r |"

!

q dx 'f

1

1

!

(for the first time this was pointed out in [1]). This being the case in the n-base region of the diode structure with prevailing carrier accumulation from the ambipolar continuity equation (1) we get d d n 0. (2) dp

This abbreviated equation may be immediately integrated to give the excess hole density distribution along the n-base region in the form P

V

i

dp

p(o) L

dp .

(3)

p(«)

Then the voltage drop across the n-base region of the structure under investigation is p(w) r p Fr = | where

E

p(0)

®B dp

D~k | L

+

dx

dp ,

P

(4)

P(w) kT b~

E = E | -{- E j-

?Mp (

bn

+ P)

da;

n

^-p dx

bn + p

(5)

Let us consider the particular model of a neutral n-base region, which contains besides donors Na , the deep compensating hole trapping centres Nt (Fig. 2) with sharply asymmetric hole and electron capture cross-sections (Opt 6'rit)•

1

AV V / V

-/

Fig. 2. Schematic energy band diagram for the n-semiconductor cointaining a single set of deep compensating hole trapping centres of density 2Vt located at energy E t as well as shallow donor impurity atoms with corresponding energy level 1: OptiV t /tp; 2: CptiVt/tp?>it; 3: r ] 0 N t f t . The dashed arrow indicates the carrier transfer between trapping levels and c-band which is negligibly small

Theory of Current-Voltage Characteristics of Semiconductor Diode Structures

65

The Poisson equation is then e

dE

4jiq

da:

=

(6)

P

where /t = 1 — /tp is the fraction of compensating deep traps occupied by electrons. Hence, the quasineutrality condition for the electron and hole concentrations yields >p +

N +

N

t

f

t v

>

dE 4jiq

(7)

dx

where N — NA — Nt is the number of uncompensated donor atoms, N J t p is the density of trapped holes, /tp=

(i+i>!t)

(8)

and pft = PUT,

0)

= pit +

rjtfVCpt

is a statistical] factor which contains besides plt = Nv exp (Ey — Et)jkT (i.e. the thermal hole density when the Fermi level is at the energy of the trap [2]), the photogeneration rate of electrons from the valence band into the deep trapping level, produced by an appropriate incident radiation with external photon flux density 0 and efficiency rj (Fig. 2). W e shall confine ourselves to the discussion of the situation when the ambipolar drift velocity is mainly defined by the ohmic relaxation of the injected space charge Q = (e/4n) dEjdx and by the modulation of the population of deep compensating traps as well. This results in _

I

b

N +

(bn+p)* I

b

q

(6+1)

N J

t v

- p ~ ( N J

[Ntp*

+

N(p

(P + Psb) P

+

y—j

2

+

t v

)

pftf-\ N(p

+

(9) pft)

where Psb = Pit

+

6 + 1

Suppose that the Dember component ED of the electrical field strength E (5) may be neglected. This allows to consider the ambipolar diffusivity _Deff in equation (2) as slowly varying function of the injected carrier densities. Consequently the term F V v [ ( d l d x ) p Y in equation (1) may be omitted. A t the same time the condition E1 i?u| means that the carrier recombination in highly-doped regions of the investigated diode structures prevails over the one in the high-resistive n-base region. This results in the boundary conditions (Fig. 1) taking the usual for non-perfectly injecting junction form [3, 4] qV*b

/ = 7(0) =

i Pn jg VP(0)

qVtp2(0)lpD

7 = I(w) = qV*n2(w)lnn 5

physica (a) 51/1

n

\

at the n + - n anode (x = 0) in n + - n - n + structures

(10a)

at the p - n anode (x = 0) in p-n-n+ structures ,

(10b)

at the n-n + cathode (x —w),

(H)

66

A . Y u . LEIDERMAN a n d P . M . KARAGEORGY-ALKALAEV

where F* a is the leakage velocity of the anodic n + - n junction (for holes) in n + - n - n + structures, F * is the leakage velocity of the anodic p + - n junction (for electrons) in p + - n - n + structures, F * is the leakage velocity of the cathodic n - n + junction (for holes) in both types of the above-mentioned structures. 3. Current—Voltage Characteristics The ohmic relaxation of the injected space charge q is the main factor defining the ambipolar drift velocity « 7

^

t

P+ :

(12)

N

which appears at lower injection levels. The corresponding free carrier distribution along the n-base region may be obtained from (3) as P -

b

P +

P(0)

l

„ -N

B (13)

p( 0)

-N

where A = p{w)

+

B =

•N

'eff

bN q (b + 1)2 •

Under these circumstances the voltage drop across the n-base region may be obtained by immediate integration of (4) (taking into account (5) and (13)) to give VT

=

+

D eff 1 Mn X

bN (b + If

N

w

p{w) -

p(0) bN q (b + If

At low injection levels when p

p(w) p(w)

Ml i']

b +

X

(14)

(14) may be reduced to I^q^NVrlw,

(15)

i.e. the current-voltage characteristics is simply Ohm's law and the resistivity of the n-base region is not disturbed by injection. At higher injection levels when p N and a quadratic rise of the anodic boundary electron density n(w) with current I (see (11)) appears, this ohmic region of the I- V characteristics is followed by the square-law region qi4(b + 1 fnB

(16)

n \

Vi

b

tfj

This square-law region of I-V dependence sufficiently differs from the well-known Lampert-Rose "ohmic-relaxation regime" I oc r p F|./w 3 [5]. I t is independent of carrier lifetime in the bulk of the base region, because in the investigated case there is no recombination effect on the excess carrier density relaxation. With increasing current density and excess hole concentration the injection modulation of the population of deep traps N t strongly affects the value of the ambipolar

Theory of Current-Voltage Characteristics of Semiconductor Diode Structures

67

drift velocity which becomes N

!

(pùy

(17)

where v& = I\q bj(b -f- l) 2 NJplThe is found to be p* +ßp_~ X(P) = Tr^-h11 2 2v* [ p (0) ßm

corresponding carrier distribution in the n-base pK~NINt — Pit NINt

ß In 2p + ß - if\Ä\ 2p(0) + ß + fiÄ\ Ì\A\ [2p(0) + ß - Ì\A\ 2p + ß + i\A\

where

ß = pitNjN tp(w) - p(w)

A = -

and

(18)

(ß2 + 4p^N/Nt)

< 0

as a consequence of the monotonie behaviour of the p(x) function. The voltage drop across the n-base region of the structure is described by the rather complicate expression D \'(NlNt) p*t,

V" (x — w) DK

jM

+ i\A\

(19)

yields and (20)

may be evaluated, where p(w) — _p(0) exp

VM IXeff,

v w

n I

From (20) it may be concluded that while immediately near the cathodic n - n + junction p(x) is a monotonically increasing function of I , in the bulk of the n-base region far enough from the cathode the carrier concentration decreases with increasing current density. This leads to a sharp increase of the voltage drop on the n-base region, which depends on the current density I nearly exponentially: VT

F0exp(

F0exp

where

D,eff qfj,p(bNtptt + 6 1

Iw b N qDeff (Ö + 1)2 plb{Ti

0 )

(21)

I 1) p{w)

P( 0)/

Bea

However, the region of the sublinearly varying 7-Fcharacteristics is restricted. Actually at high injection levels, when p exceeds pSb, trapping levels are totally occupied by holes and therefore the ambipolar drift velocity decreases with increasing p as *

b — l . q (b + l)2 p2

(22)

Consequently the I-V characteristics is again given by a square law defined by expression (16) with N replaced by Nt.

A. Yu. Leiderman and P. M. Kabageorgy-Alkalaev

68

The voltage drop VT exponentially increasing with the ambipolar drift velocity n ~ v J i I , p S b ( T , i>)] (and hence with the current density I ) within the sufficiently extended sublinear region of I-V dependence is the most remarkable feature of the investigated diode structures with strong carrier accumulation. 1 ) Moreover at the injection levels corresponding to the sublinear region of the I-V dependence the ambipolar drift velocity v

T_ —

V. = Va =

b_

7

Nt

; • 1)'"

(h

7

A ,, exp

I —

Cpt

-

b + 1

N,

sufficiently depends on the various external influences changing the number of trapped holes, as to heating, photoexcitation, pressure etc. Therefore, the investigated structures with strong accumulation within the sublinear regions of the I-V dependence are extremely sensitive to the variation of incident radiation and temperature. Moreover this sensitivity sufficiently exceeds the sensitivity of the adjacent ohmic and square-law regions of the I-V curve. 4. Sensitivity of Diode Structures with Strong Carrier Accumulation to Incident Radiation and Temperature Variation Let us consider t h e small voltage |AF