- Email: [email protected]
Longitudinal stark-cyclotron waves in semiconductor with a superlattice
Solid State Communications, Vol. 39, pp. 843-845. Pergamon Press Ltd. 1981. Printed in Great Britain.
0038-1098/81 [310843-03502.00•0
LONGITUDINAL STARK-CYCLOTRON WAVES IN SEMICONDUCTOR WITH A SUPERLATTICE A.P. Tetervov Institute of Semiconductor Physics, Academy of Sciences of the lithuanian SSR, Vilnius, USSR (Received 12 December 1980 by E.A. Kaner)
Semiconductor with a superlattice in static electrical and magnetic fields parallel to the superlattice axis is considered. The existence of Starkcyclotron longitudinal waves in non-local approximation is predicted. AN ELECTRON in the narrow-band semiconductor under the action of the external electric field Eo is known to perform the periodic motion with Stark frequency ~2 = (e/h)(Eoa) with e and a standing for the charge of carrier and crystallic lattice constant respectively. The field must be strong enough a>>v
(1)
where v is the collision frequency, or otherwise the electron is withdrawn from the finite motion trajectory through the scattering. Since in standard semiconductors a is of atomic dimensions the above inequality fulfils at very strong fields only. At the same time semiconductor with a superlattice [1,2] is characterized by an additional periodic potential with period d much greater than a. Due to that fact condition e
(2)
ws = -~ Eod >>v
is feasible at really achievable field strength. When static magnetic field Ho is applied in addition to the electrical field an electron performs periodic motion too with cyclotron frequency ~oc = eHo/mo c where mo is the electron effective mass. (The relation 6% >> v is supposed to be fulfilled also.) Since there exist two types of finite motion of the electron in the above situation one can expect the appearance of certain peculiarities in expression for the static or high-frequency current. From general considerations it is clear that these peculiarities become most apparent near the Stark-cyclotron resonancestatic, when 608 = coe, or dynamical, when the incident electromagnetic wave frequency co satisfies the condition w = lw s+m¢o c
(l,m = 0 , + 1 . . . . ).
superlattice only since the finite motion of an electron under the action of electrical field is possible only in such materials. For the first time the possibility of the Starkcyclotron resonance in semiconductor with a superlattice was pointed out by Bass et al. [4]. However in order to couple two mentioned finite motions the authors of [4] made use of special fields geometry: electrical field was along the superlattice axis and magnetic - at small angle to it. When both fields are parallel to the superlattice axis Stark-cyclotron resonance is absent. In the present paper we want to pay attention to the possibility of existence of the Stark-cyclotron resonance and associated phenomena. When external fields are parallel to the super•attice axis. In order to get this one must take into account the spatial dispersion. Thus the coupling of electron periodic motions in static fields can occur not only owing to fields geometry but also with allowance for spatial inhomogeneities. The semiconductor with one-dimensional superlattice with static electrical and magnetic fields along its axis is considered. In the one-miniband approximation the electron energy spectrum will be described by 1
2
pxd
e(p) = ~mo ( p ' + p ~ ) - A cos T
(4)
with py, Pz and A standing respectively for the transverse momentum and miniband width. Besides, we suppose the conditions of the electron quasi-classical motion, e.g. h - l A >>~ s , ~Oc, v to be fulfilled. These relations permit to use classical kinetic equation. Using pondermotive equation
(3)
In the latter case one can expect the excitation of Starkcyclotron waves by analogy with excitation of cyclotron waves in solids [3]. It is necessary to emphasize the fact that Starkcyclotron wave can occur in semiconductor with a
dt
e Eo + c v x Ho
(5)
one can get for the electron velocity v = Oe(p)/ap vlt) = I . .
sm | k
V±o sin (~ -- ¢oet)l ) 843
h
d + c o s t J];
v±o cos (a -- ¢%0; (6)
LONGITUDINAL STARK-CYCLOTRON WAVES IN SEMICONDUCTOR
844 with
m
*
tl2
_
2
Ad2'
=
Vto
v 2'',y~u)
tgd
-
v~ (0)
vy(0)"
High-frequency conductivity tensor with allowance for the spatial dispersion, is obtained using the standard procedure (see, e.g. [5] ). Substituting the obtained expressions into Maxwell equations one can get the spectra of electromagnetic waves propagating in the semiconductor with a superlattice plasma. In general case the dispersion equations of these waves are too cumbersome. We shall consider the relatively simple but sufficiently illustrative at the same time case of the longitudinal wave propagation. The general dispersion equation of longitudinal waves (see [5])
klkje~i(co, k) = 0
(7)
with wave vector k taken in form k = {kx;ky ;0} can be written as follows
eo + ~,4*r [%x cos2~0 + % sin ~0 cos ¢ + oyy sin2~o]
Vol. 39, No. 7
function with x = kx (A//Eo); Im (y) is the modified Bessel function with y = [ky (VTh/coc)] z ;VTrh is the thermal electron velocity;/3 = A/moV2h. One can see easily from equation (9) that at x = y = 0 the coupled Stark-cyclotron oscillations do not arise. Thus, in our case these waves exist only with allowance for the spatial inhomogenity parameters x and y. Physically they imply respectively ratio of miniband width to the energy accumulated by electron at wavelength along the superlattice axis in electrical field and square ratio of the electron Larmour radius to the cyclotron frequency. The latter is well-known in the theory of magnetized plasma (gaseous [5] and solid-state [3]). The former parameter is bound up with an electron energy spectrum (for the first time it was introduced in [6] ). In order to solve the equation (8) we notice that according to the relation (2) the main contribution into the sum over n is determined by n = 0, -+ 1. In weak spatial dispersion over transverse wave number limit we expand the Bessel function Im (y) into a power series in y ~ 1 [7], thus the equation (8) takes the form:
CO
= 0
(8)
eo ~ cog" sin2~ ~_ ¢.,0
with eo, lattice dielectric constant, ~0, the angle between superlattice axis and k. The conductivity tensor components in equation (8) have the form
+ i~ -1
+ ~.
lnatmn(x,y),
I , rrt, rl = - o o
x
(l x
=
Oxy+OYx
x t
--ico2II
(/3 m*tl/2mO-]y - I
+ i),~
~
J
x
t,o
+
+iX~ = d J ? ( x ) ]
ma ao,,,(x,y),
(9)
l , rrt~ rl = - o °
where the following designations are introduced:
dx
-- +/3x
(x)] dx ]
(10)
-= c o l ,-11 + - c o l ,-1 - 1 ' ~ =
V/COs"
One can see easily that in the Stark-cyclotron resonance region coo = pco, _+ coe the solution of equation (10) may be written in the form co(k) = coo + 7t + iwl with 71
~
coo±
2eowO
JZp(x)
1 + 2p -
1 + 2X~ -
alr,m(x,Y) = ( _ i)n exp ( - - y ) 1,(/3)Jt(x)Jt+,,(x) Im (y)
4n
lo (/3)
[4ne2N~ ~/2
coo,=
m* : '
col, m
v
"Y + incos
= (47re2Ntl'2
coo±
2 eo coo cos
dx
~ 1 -+ 2p
m-T-:
is the "longitudinal" and "transverse" plasma frequencies; cot, m = co + lcos +mcoc + iv;Jp(x) is the Bessel
]
t
whereX=I~(~)/Io(/3),S2~.l
m(lx-'+in/3-')azmn(x,y),
|, ln, rl= -oo
o,~ = ico2±y-'
dx
m*g2t,+x) [J?(x) 0 --2X~
\
=
J~(x)
2mo
OO
Oxx = --co~± x-'
7r~2 ctg2~o 2
l=-oo
Obviously there exists the possibility of longitudinal
Vol. 39, No. 7
84
LONGITUDINAL STARK-CYCLOTRON WAVES IN SEMICONDUCTOR
Stark-cyclotron waves instability (tO'~'> O) in dense enough plasma of the semiconductor with a superlattice. In particular at x < 1 and p = -+ 1 these oscillations are unstable under the following condition tOg± v ptO,+tO° -sin2~.o. 4Co <-+ tOo tos tO, tOc
3`2~
togL
\
/ tO~'~--7
(13) x
(Lattice temperature is supposed to be low enough so that/3>> 1) For the numerical estimations one can take the following parameters: A = 0.1 eV, d = 2 x 10 -6 cm, Eo = 1.5 kV cm -I ,Ho = 3 x 10 3 0 e , m o = 10 -29 g, 7)Th = 5 X 10 7 cm sec -~, 3' = 1012 sec-1 • At such values tOe ~ tOs = 5 X 1012 sec -! >> 3'. Then conditions x = 1 and y = 1 correspond respectively to the wave number values k~ = 1.5 x 104 cm -I and ky = l0 s cm -l . Substituting these values into equation (13) one may be convinced of its correctness when for example eo ~ 10, x = 0, 1 and density of electrons in the miniband is equal to N = 10 Is cm -3. In the opposite limit of strong spatial dispersion the Bessel function Im 0 ' ) in equations (8) and (9) can be replaced by its asymptotic expression for large argument value (y >> 1)(see [7] ). It can be shown that solution of the equation (I0) near the Stark-cyclotron resonance tOo = ptO, + rtOe (r 2 ,~y) for the applicability of mentioned expansion) after some tedious but straightforward calculations takes the form tO = 600 + 3,2 + itO~ with
~(2ffy3)-1/2 J ~ ( X ) [ 1
COtOe
+2~
~@~ sin2~o(14
X/
\-1,2
1 eOtOg'tO~2~y3) 2c )~1n20
(,~~-x]
tO _ _ / 3 ] ~ ] .
(15
One can see using the above numerical estimations that tO~ is negative through the small parameter y-3/2 1 Hence the longitudinal Stark-cyclotron oscillations in short-wavelength limit are weakly damped. Acknowledgements - The author is indebted to Prof. F.G. Bass and Prof. E.A. Kaner for their attention and many fruitful discussions. REFERENCES 1. 2. 3. 4. 5. 6.
L. E s a k i & R . Tsu, I B M Z R e s . Dev. 14,61 (19701 S.V. Gaponov, B.M. Luskin & N.N. Salashenko, Fiz. Tekh. Poluprov. 14, 1468 (1980). E.A. Kaner & V.G. Skobov, Adv. Phys. 17, N69, 605 (1968). F.G. Bass, V.V. Zorchenko & V.I. Shashora,/~'s~ V2ETF, 31,345 (1980) V.D. Shafranov, in Voprosy teoril plazmy, Issue Atomizdat, Moscow (1963). A. Erdelyi et aL , H~gher Transcendental Function Vol. 2. McGraw-Hill, New York (1953).
0038-1098/81 [310843-03502.00•0
LONGITUDINAL STARK-CYCLOTRON WAVES IN SEMICONDUCTOR WITH A SUPERLATTICE A.P. Tetervov Institute of Semiconductor Physics, Academy of Sciences of the lithuanian SSR, Vilnius, USSR (Received 12 December 1980 by E.A. Kaner)
Semiconductor with a superlattice in static electrical and magnetic fields parallel to the superlattice axis is considered. The existence of Starkcyclotron longitudinal waves in non-local approximation is predicted. AN ELECTRON in the narrow-band semiconductor under the action of the external electric field Eo is known to perform the periodic motion with Stark frequency ~2 = (e/h)(Eoa) with e and a standing for the charge of carrier and crystallic lattice constant respectively. The field must be strong enough a>>v
(1)
where v is the collision frequency, or otherwise the electron is withdrawn from the finite motion trajectory through the scattering. Since in standard semiconductors a is of atomic dimensions the above inequality fulfils at very strong fields only. At the same time semiconductor with a superlattice [1,2] is characterized by an additional periodic potential with period d much greater than a. Due to that fact condition e
(2)
ws = -~ Eod >>v
is feasible at really achievable field strength. When static magnetic field Ho is applied in addition to the electrical field an electron performs periodic motion too with cyclotron frequency ~oc = eHo/mo c where mo is the electron effective mass. (The relation 6% >> v is supposed to be fulfilled also.) Since there exist two types of finite motion of the electron in the above situation one can expect the appearance of certain peculiarities in expression for the static or high-frequency current. From general considerations it is clear that these peculiarities become most apparent near the Stark-cyclotron resonancestatic, when 608 = coe, or dynamical, when the incident electromagnetic wave frequency co satisfies the condition w = lw s+m¢o c
(l,m = 0 , + 1 . . . . ).
superlattice only since the finite motion of an electron under the action of electrical field is possible only in such materials. For the first time the possibility of the Starkcyclotron resonance in semiconductor with a superlattice was pointed out by Bass et al. [4]. However in order to couple two mentioned finite motions the authors of [4] made use of special fields geometry: electrical field was along the superlattice axis and magnetic - at small angle to it. When both fields are parallel to the superlattice axis Stark-cyclotron resonance is absent. In the present paper we want to pay attention to the possibility of existence of the Stark-cyclotron resonance and associated phenomena. When external fields are parallel to the super•attice axis. In order to get this one must take into account the spatial dispersion. Thus the coupling of electron periodic motions in static fields can occur not only owing to fields geometry but also with allowance for spatial inhomogeneities. The semiconductor with one-dimensional superlattice with static electrical and magnetic fields along its axis is considered. In the one-miniband approximation the electron energy spectrum will be described by 1
2
pxd
e(p) = ~mo ( p ' + p ~ ) - A cos T
(4)
with py, Pz and A standing respectively for the transverse momentum and miniband width. Besides, we suppose the conditions of the electron quasi-classical motion, e.g. h - l A >>~ s , ~Oc, v to be fulfilled. These relations permit to use classical kinetic equation. Using pondermotive equation
(3)
In the latter case one can expect the excitation of Starkcyclotron waves by analogy with excitation of cyclotron waves in solids [3]. It is necessary to emphasize the fact that Starkcyclotron wave can occur in semiconductor with a
dt
e Eo + c v x Ho
(5)
one can get for the electron velocity v = Oe(p)/ap vlt) = I . .
sm | k
V±o sin (~ -- ¢oet)l ) 843
h
d + c o s t J];
v±o cos (a -- ¢%0; (6)
LONGITUDINAL STARK-CYCLOTRON WAVES IN SEMICONDUCTOR
844 with
m
*
tl2
_
2
Ad2'
=
Vto
v 2'',y~u)
tgd
-
v~ (0)
vy(0)"
High-frequency conductivity tensor with allowance for the spatial dispersion, is obtained using the standard procedure (see, e.g. [5] ). Substituting the obtained expressions into Maxwell equations one can get the spectra of electromagnetic waves propagating in the semiconductor with a superlattice plasma. In general case the dispersion equations of these waves are too cumbersome. We shall consider the relatively simple but sufficiently illustrative at the same time case of the longitudinal wave propagation. The general dispersion equation of longitudinal waves (see [5])
klkje~i(co, k) = 0
(7)
with wave vector k taken in form k = {kx;ky ;0} can be written as follows
eo + ~,4*r [%x cos2~0 + % sin ~0 cos ¢ + oyy sin2~o]
Vol. 39, No. 7
function with x = kx (A//Eo); Im (y) is the modified Bessel function with y = [ky (VTh/coc)] z ;VTrh is the thermal electron velocity;/3 = A/moV2h. One can see easily from equation (9) that at x = y = 0 the coupled Stark-cyclotron oscillations do not arise. Thus, in our case these waves exist only with allowance for the spatial inhomogenity parameters x and y. Physically they imply respectively ratio of miniband width to the energy accumulated by electron at wavelength along the superlattice axis in electrical field and square ratio of the electron Larmour radius to the cyclotron frequency. The latter is well-known in the theory of magnetized plasma (gaseous [5] and solid-state [3]). The former parameter is bound up with an electron energy spectrum (for the first time it was introduced in [6] ). In order to solve the equation (8) we notice that according to the relation (2) the main contribution into the sum over n is determined by n = 0, -+ 1. In weak spatial dispersion over transverse wave number limit we expand the Bessel function Im (y) into a power series in y ~ 1 [7], thus the equation (8) takes the form:
CO
= 0
(8)
eo ~ cog" sin2~ ~_ ¢.,0
with eo, lattice dielectric constant, ~0, the angle between superlattice axis and k. The conductivity tensor components in equation (8) have the form
+ i~ -1
+ ~.
lnatmn(x,y),
I , rrt, rl = - o o
x
(l x
=
Oxy+OYx
x t
--ico2II
(/3 m*tl/2mO-]y - I
+ i),~
~
J
x
t,o
+
+iX~ = d J ? ( x ) ]
ma ao,,,(x,y),
(9)
l , rrt~ rl = - o °
where the following designations are introduced:
dx
-- +/3x
(x)] dx ]
(10)
-= c o l ,-11 + - c o l ,-1 - 1 ' ~ =
V/COs"
One can see easily that in the Stark-cyclotron resonance region coo = pco, _+ coe the solution of equation (10) may be written in the form co(k) = coo + 7t + iwl with 71
~
coo±
2eowO
JZp(x)
1 + 2p -
1 + 2X~ -
alr,m(x,Y) = ( _ i)n exp ( - - y ) 1,(/3)Jt(x)Jt+,,(x) Im (y)
4n
lo (/3)
[4ne2N~ ~/2
coo,=
m* : '
col, m
v
"Y + incos
= (47re2Ntl'2
coo±
2 eo coo cos
dx
~ 1 -+ 2p
m-T-:
is the "longitudinal" and "transverse" plasma frequencies; cot, m = co + lcos +mcoc + iv;Jp(x) is the Bessel
]
t
whereX=I~(~)/Io(/3),S2~.l
m(lx-'+in/3-')azmn(x,y),
|, ln, rl= -oo
o,~ = ico2±y-'
dx
m*g2t,+x) [J?(x) 0 --2X~
\
=
J~(x)
2mo
OO
Oxx = --co~± x-'
7r~2 ctg2~o 2
l=-oo
Obviously there exists the possibility of longitudinal
Vol. 39, No. 7
84
LONGITUDINAL STARK-CYCLOTRON WAVES IN SEMICONDUCTOR
Stark-cyclotron waves instability (tO'~'> O) in dense enough plasma of the semiconductor with a superlattice. In particular at x < 1 and p = -+ 1 these oscillations are unstable under the following condition tOg± v ptO,+tO° -sin2~.o. 4Co <-+ tOo tos tO, tOc
3`2~
togL
\
/ tO~'~--7
(13) x
(Lattice temperature is supposed to be low enough so that/3>> 1) For the numerical estimations one can take the following parameters: A = 0.1 eV, d = 2 x 10 -6 cm, Eo = 1.5 kV cm -I ,Ho = 3 x 10 3 0 e , m o = 10 -29 g, 7)Th = 5 X 10 7 cm sec -~, 3' = 1012 sec-1 • At such values tOe ~ tOs = 5 X 1012 sec -! >> 3'. Then conditions x = 1 and y = 1 correspond respectively to the wave number values k~ = 1.5 x 104 cm -I and ky = l0 s cm -l . Substituting these values into equation (13) one may be convinced of its correctness when for example eo ~ 10, x = 0, 1 and density of electrons in the miniband is equal to N = 10 Is cm -3. In the opposite limit of strong spatial dispersion the Bessel function Im 0 ' ) in equations (8) and (9) can be replaced by its asymptotic expression for large argument value (y >> 1)(see [7] ). It can be shown that solution of the equation (I0) near the Stark-cyclotron resonance tOo = ptO, + rtOe (r 2 ,~y) for the applicability of mentioned expansion) after some tedious but straightforward calculations takes the form tO = 600 + 3,2 + itO~ with
~(2ffy3)-1/2 J ~ ( X ) [ 1
COtOe
+2~
~@~ sin2~o(14
X/
\-1,2
1 eOtOg'tO~2~y3) 2c )~1n20
(,~~-x]
tO _ _ / 3 ] ~ ] .
(15
One can see using the above numerical estimations that tO~ is negative through the small parameter y-3/2 1 Hence the longitudinal Stark-cyclotron oscillations in short-wavelength limit are weakly damped. Acknowledgements - The author is indebted to Prof. F.G. Bass and Prof. E.A. Kaner for their attention and many fruitful discussions. REFERENCES 1. 2. 3. 4. 5. 6.
L. E s a k i & R . Tsu, I B M Z R e s . Dev. 14,61 (19701 S.V. Gaponov, B.M. Luskin & N.N. Salashenko, Fiz. Tekh. Poluprov. 14, 1468 (1980). E.A. Kaner & V.G. Skobov, Adv. Phys. 17, N69, 605 (1968). F.G. Bass, V.V. Zorchenko & V.I. Shashora,/~'s~ V2ETF, 31,345 (1980) V.D. Shafranov, in Voprosy teoril plazmy, Issue Atomizdat, Moscow (1963). A. Erdelyi et aL , H~gher Transcendental Function Vol. 2. McGraw-Hill, New York (1953).