Abstract

A quantitative derivation is presented for the production of the photoacoustic signal in semiconductors, taking into account finite carrier diffusion and recombination times in the solid.

© 1982 Optical Society of America

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References

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  1. A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
    [CrossRef]
  2. L. C. Aamodt, J. C. Murphy, J. G. Parker, J. Appl. Phys. 48, 927 (1977).
    [CrossRef]
  3. L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
    [CrossRef]
  4. F. A. McDonald, G. C. Wentsel, J. Appl. Phys. 49, 2313 (1978).
    [CrossRef]
  5. H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
    [CrossRef]
  6. C. L. Cesar, H. Vargas, J. A. Meyer, L. C. M. Miranda, Phys. Rev. Lett. 42, 1560 (1979).
    [CrossRef]
  7. W. Shockley, W. T. Read, Phys. Rev. 87, 835 (1952).
    [CrossRef]
  8. R. N. Hall, Phys. Rev. 87, 387 (1952).
    [CrossRef]
  9. S. Wang, G. Wallis, Phys. Rev. 105, 1459 (1957).
    [CrossRef]
  10. W. Van Roosbroeck, W. Shockley, Phys. Rev. 94, 1558 (1954).
    [CrossRef]
  11. W. P. Dumke, Phys. Rev. 105, 139 (1957).
    [CrossRef]

1979

C. L. Cesar, H. Vargas, J. A. Meyer, L. C. M. Miranda, Phys. Rev. Lett. 42, 1560 (1979).
[CrossRef]

1978

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

F. A. McDonald, G. C. Wentsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

1977

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. G. Parker, J. Appl. Phys. 48, 927 (1977).
[CrossRef]

1976

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

1957

S. Wang, G. Wallis, Phys. Rev. 105, 1459 (1957).
[CrossRef]

W. P. Dumke, Phys. Rev. 105, 139 (1957).
[CrossRef]

1954

W. Van Roosbroeck, W. Shockley, Phys. Rev. 94, 1558 (1954).
[CrossRef]

1952

W. Shockley, W. T. Read, Phys. Rev. 87, 835 (1952).
[CrossRef]

R. N. Hall, Phys. Rev. 87, 387 (1952).
[CrossRef]

Aamodt, L. C.

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. G. Parker, J. Appl. Phys. 48, 927 (1977).
[CrossRef]

Bennett, H. S.

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

Cesar, C. L.

C. L. Cesar, H. Vargas, J. A. Meyer, L. C. M. Miranda, Phys. Rev. Lett. 42, 1560 (1979).
[CrossRef]

Dumke, W. P.

W. P. Dumke, Phys. Rev. 105, 139 (1957).
[CrossRef]

Forman, R. A.

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

Gersho, A.

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

Hall, R. N.

R. N. Hall, Phys. Rev. 87, 387 (1952).
[CrossRef]

McDonald, F. A.

F. A. McDonald, G. C. Wentsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

Meyer, J. A.

C. L. Cesar, H. Vargas, J. A. Meyer, L. C. M. Miranda, Phys. Rev. Lett. 42, 1560 (1979).
[CrossRef]

Miranda, L. C. M.

C. L. Cesar, H. Vargas, J. A. Meyer, L. C. M. Miranda, Phys. Rev. Lett. 42, 1560 (1979).
[CrossRef]

Murphy, J. C.

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. G. Parker, J. Appl. Phys. 48, 927 (1977).
[CrossRef]

Parker, J. G.

L. C. Aamodt, J. C. Murphy, J. G. Parker, J. Appl. Phys. 48, 927 (1977).
[CrossRef]

Read, W. T.

W. Shockley, W. T. Read, Phys. Rev. 87, 835 (1952).
[CrossRef]

Rosencwaig, A.

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

Shockley, W.

W. Van Roosbroeck, W. Shockley, Phys. Rev. 94, 1558 (1954).
[CrossRef]

W. Shockley, W. T. Read, Phys. Rev. 87, 835 (1952).
[CrossRef]

Van Roosbroeck, W.

W. Van Roosbroeck, W. Shockley, Phys. Rev. 94, 1558 (1954).
[CrossRef]

Vargas, H.

C. L. Cesar, H. Vargas, J. A. Meyer, L. C. M. Miranda, Phys. Rev. Lett. 42, 1560 (1979).
[CrossRef]

Wallis, G.

S. Wang, G. Wallis, Phys. Rev. 105, 1459 (1957).
[CrossRef]

Wang, S.

S. Wang, G. Wallis, Phys. Rev. 105, 1459 (1957).
[CrossRef]

Wentsel, G. C.

F. A. McDonald, G. C. Wentsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

J. Appl. Phys.

A. Rosencwaig, A. Gersho, J. Appl. Phys. 47, 64 (1976).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. G. Parker, J. Appl. Phys. 48, 927 (1977).
[CrossRef]

L. C. Aamodt, J. C. Murphy, J. Appl. Phys. 49, 3036 (1978).
[CrossRef]

F. A. McDonald, G. C. Wentsel, J. Appl. Phys. 49, 2313 (1978).
[CrossRef]

H. S. Bennett, R. A. Forman, J. Appl. Phys. 48, 1432 (1977).
[CrossRef]

Phys. Rev.

W. Shockley, W. T. Read, Phys. Rev. 87, 835 (1952).
[CrossRef]

R. N. Hall, Phys. Rev. 87, 387 (1952).
[CrossRef]

S. Wang, G. Wallis, Phys. Rev. 105, 1459 (1957).
[CrossRef]

W. Van Roosbroeck, W. Shockley, Phys. Rev. 94, 1558 (1954).
[CrossRef]

W. P. Dumke, Phys. Rev. 105, 139 (1957).
[CrossRef]

Phys. Rev. Lett.

C. L. Cesar, H. Vargas, J. A. Meyer, L. C. M. Miranda, Phys. Rev. Lett. 42, 1560 (1979).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic configuration of the photoacoustic cell.

Fig. 2
Fig. 2

Frequency dependence of the pressure fluctuation obtained from Eqs. (15) and (17) for a typical thermally thick (l = 1 cm) Si sample with a carrier recombination time of 1 msec.

Fig. 3
Fig. 3

Frequency dependence of the pressure fluctuation obtained from Eqs. (15) and (23) for a typical thermally thin (l = 1 μm) Si sample with a carrier recombination time of 1 msec.

Equations (30)

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2 ϕ s x 2 = 1 α s ϕ s t - Q s ( x , t ) K s ,             - l < x < 0 ,
Q s = h ν n t ( x , t ) ,
n t ( x , t ) = β I 0 h ν exp ( β x ) exp ( j ω t ) .
n t = - n τ + D 2 n x 2 + β I 0 h ν exp ( β x ) exp ( j ω t ) ,             - l < x < 0.
- D n x ) x = 0 = v s n ( 0 , t ) ,
D n x ) x = - l = v s n ( - l , t )
n ( x , t ) = β I 0 h ν D ( β 2 - α 2 ) × [ χ 1 exp ( α x ) + χ 2 exp ( - α x ) - exp ( β x ) ] exp ( j ω t ) ,
χ 1 = ( 1 + v s v d ) ( β α + v s v d ) exp ( α l ) - ( 1 - v s v d ) ( β α - v s v d ) exp ( - β l ) ( 1 + v s v d ) 2 exp ( α l ) - ( 1 - v s v d ) 2 exp ( - α l ) ,
χ 2 = ( 1 - v s v d ) ( β α + v s v d ) exp ( - α l ) - ( 1 + v s v d ) ( β α - v s v d ) exp ( - β l ) ( 1 + v s v d ) 2 exp ( α l ) - ( 1 - v s v d ) 2 exp ( - α l ) .
Q s = j w β I 0 D ( β 2 - α 2 ) [ χ 1 exp ( α x ) + χ 2 exp ( - α x ) - exp ( β x ) ] exp ( j ω t ) .
2 ϕ g x 2 = 1 α g ϕ g t ,             0 < x < l g ,
2 ϕ g x 2 = 1 α b ϕ b t ,             - ( l + l b ) < x < - l .
ϕ s ( x , t ) = [ U exp ( σ s x ) + V exp ( - σ s x ) + B exp ( β x ) + C 1 exp ( α x ) + C 2 exp ( - α x ) ] exp ( j ω t ) ,             - l < x < 0.
ϕ g ( x , t ) = θ exp ( - σ g x ) exp ( j ω t ) ,             0 < x < l g ,
ϕ b ( x , t ) = W exp ( σ b x ) exp ( j ω t )             - l - l b < x < - l ,
σ i = ( 1 + j ) a i ,
B = j w β I 0 k s ( β 2 - σ s 2 ) D ( β 2 - α 2 ) ,
C 1 = - j w β I 0 k s ( α 2 - σ s 2 ) D ( β 2 - α 2 ) χ 1 ,
C 2 = j w β I 0 k s ( α 2 - σ s 2 ) D ( β 2 - α 2 ) χ 2 .
θ = { [ ( 1 - b ) ( r + 1 ) exp ( - l σ s ) + ( 1 + b ) ( r - 1 ) exp ( l σ s ) - 2 ( r - b ) exp ( - β l ) ] B + [ ( 1 - b ) ( λ + 1 ) exp ( - l σ s ) + ( 1 + b ) ( λ - 1 ) exp ( l σ s ) - 2 ( λ - b ) exp ( - α l ) ] C - [ ( 1 - b ) ( λ - 1 ) exp ( - l σ s ) + ( 1 + b ) ( λ + 1 ) exp ( l σ s ) - 2 ( λ + b ) exp ( α l ) ] C 2 } × [ ( 1 - b ) ( 1 - g ) exp ( - l σ s ) - ( 1 + b ) ( 1 + g ) exp ( l σ s ) ] - 1
δ P ( t ) = Q exp j ( ω t - π / 4 ) ,             Q = γ P 0 θ 2 l g T 0 a g ,
C 1 j ω β 2 I 0 k s ( β 2 - α 2 ) D α ( α 2 - σ s 2 ) exp ( l α ) - exp ( - β l ) exp ( l α ) - exp ( - l α ) ,
C 2 j ω β 2 I 0 k s ( β 2 - α 2 ) D α ( α 2 - σ s 2 ) exp ( - l α ) - exp ( - β l ) exp ( l α ) - exp ( - l α ) .
θ j ω I 0 k s D α ( α 2 - σ s 2 ) [ λ - exp ( l α ) + exp ( - l α ) exp ( l α ) - exp ( - l α ) ] .
α σ s = ( 1 + j f f 2 ) 1 / 2 ( 1 + j ) ( f 1 f ) 1 / 2 = { 1 for f f 1 , 1 for f f 1 .
δ P = γ P 0 I 0 α g 1 / 2 α s 1 / 2 τ l g T 0 k s exp ( j ω t ) ,
δ P = γ P 0 I 0 α g 1 / 2 α s τ 1 / 2 exp [ j ( ω t - π / 4 ) ] 2 π g T 0 k s D 1 / 2 f 1 / 2 ( 1 + j f f 2 ) 1 / 2 . { 1 ,             l α 1 , 1 l α ,             l α 1 ,
γ j ω I 0 σ s k b σ b D α ( α 2 - σ s 2 ) [ λ + 2 b exp ( l α ) - exp ( - l α ) - ( l σ s + b ) × exp ( l α ) + exp ( - l α ) exp ( l α ) - exp ( - l α ) ] .
δ P = γ P 0 I 0 α g 1 / 2 α b 1 / 2 τ l g T 0 k b exp ( j ω t ) .
δ P = γ P 0 I 0 α g 1 / 2 α b 1 / 2 τ l g T 0 k b ( 1 + j f f 2 ) exp ( j ω t ) ,

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