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Sequential tunneling of holes in p-type semiconductor double-barrier heterostructures
Surface Science 267 (1992) 409-412 North-Holland
surface science
Sequential tunneling of holes in p-type semiconductor double-barrier heterostructures E.C. Valadares, F.W. S h e a r d and L. Eaves Department of Physics, Unicersityof Nottingham, Nottingham, NG7 2RD, UK Received 7 June 1991" accepted for publication 31 July 1991
We propose an extension of Bardeen's transfer Hamiltonian formalism to treat the sequential tunneling of holes in p-type semiconductor heterostructures displaying weakly coupled QW's. Our discussion is aimed at modeling recent resonant magnetotunneling experiments on double-barrier heterostructures displaying an emitter accumulation layer and wide barriers.
Recent resonant magnetotunneling spectroscopy (RMTS) experiments [1] have revealed important features of the valence band of p-type quantum-well heterostructures, such as the anticrossing and negative mass behaviour which arises from the spin-orbit term in the Luttinger Hamiltonian [2]. These and previous transport and optical experiments on G a A s / A I A s superlattices [3,4] strongly support the sequential tunneling of holes in such heterostructures. A formalism incorporating the sequential character of tunneling is thus of great interest for the discussion of such experiments. Traditionally, Bardeen's transfer Hamiltonian formalism has been used to model sequential tunneling of electrons (one-band systems) in n-type semiconductor heteros,ructures [5]. In the present paper, we outline an extension of this formalism to treat tunneling of holes in multiband systems within the effective-mass approximation. In order to discuss Bardeen's formalism in the context of holes, we consider the four-band Luttinger effective-mass approximation and neglect the spin-orbit split-off band. A further simplification which can be introduced is the axial (or cylindrical) approximation [6]. Under this approximation, the 4 × 4 Luttinger Hamiltonian can be diagonalized into 2 × 2 blocks by means of a suitable unitary transformation [6]. The formal-
ism can treat the complete Luttinger Hamiltonian, but these simplifications save considerable computing time without losing the main feature of band mixing between heavy holes (HH) and light holes (LH). Nevertheless, the split-off band (A = 340 meV) has to be included if states with comparable energies are considered. The Luttinger Hamiltonian can then be written [2], HL(k ) = a k 2 + /3( k . L) 2 + V( z),
(1)
where L = ( L , , L v,L_) are the 4 × 4 angular 3 momentum matrices for quantum number L and V(z) is the heterostructure potential. As discussed previously [7], one can define a current density operator J related to the Luttinger Hamiltonian. For our purposes, only the z-col.~ponent of J, J~, is relevant. It is used to obtain the solutions of the Schr6dinger equation for H L, which involve the matching of the wave-function and the flux at each interface [2]. The matrix expression for J. associated with H L reads [7] _
1
(J:)ij,= ~(D~. ~, + D~.)k~,
(2)
where D is the inverse effective-mass tensor asso3 ciated with eq. (1) and j,j' refer to the spin basis states (3, ~), /3 ~ {x, y, z} and _ 2I' 5~ , . the sum over repeated indices is implied. This operator is required to calculate tunneling cur-
0039-6028/92/$05.00 © 1992 - Elsevier Science Publishers B.V. and Yamada Science Foundation. All rights reserved
410
E.C Valadares et ai. / Sequential tunneling of holes
rents through Bardeen's formalism. The essential idea of this formalism is to treat tunneling in terms of transitions from left-hand (L) and righthand (R) states. The transition rate WLR between subsystems L and R then reads 2~" WLR =
7// I MLR 126(E L - ER) ,
(3)
where (4)
MLR = ihJLR~kllt,kilR ,
and k II refers to the transverse components of the wave-vector in the R and L subsystems. For electrons, the expression for /LR is
-ih (d~L ~ ~ * -
JLR= 2m 8
dz
R
~L
dqt~) dz :=, ,
(5)
where m B is the effective mass in a region (barrier) common to the R and L subsystems in which expression (5) is evaluated ( z = s ) . This expression has the same form as the probability flux carried by a single wave-function (see, e.g., ref. [8]). From expressions (3) and (4) it is clear that conservation of energy and in-plane momentum, k II = (kx, k:), during tunneling is required. The corresponding expression for JLR for the Luttinger Hamiltonian, which involves multibands, is obtained in the same way as for electrons, since Bardeen's formalism essentially corresponds to a first-order timz-dependent perturbation theory. It reads 1
JLR = -- ~ih(qt~Jflt L - q t L J : ~ ) z= ~,
(6)
where now qt R and qr L refer to four-component (or six-component, if the split-off band is included) wave-functions, which are solutions of the Schr6dinger equation for subsystems R and L, respectively. Although the application of the formalism to holes invoives essentially the same steps used for electrons, there are some important differences which arise from the mixing between heavy- and light-hole states. For systems involving the transfer of holes from one quantum well to another, the application of the formalism is quite similar to its electronic counterpart. Time-dependent models of tunneling of electrons were primarily developed
to calculate the transition rates between quasibound L and R eigenstates, such as 2DEG formed in the emitter accumulation layer of n-type single- and double-barrier heterostructures and states in the QW. Such states carry no current in the direction of confinement and, therefore, cannot be treated using a time-independent formalism [9]. We address our discussion to the p-type GaAs/A1As double-barrier structures studied recently by RMTS [1]. They comprise a 2DHG in the emitter accumulation layer and QW states which are probed by the RMTS. Since these experimei3ts provide the dispersion curves of the QW, we expect the tunneling current to be primarily determined by the transfer of charge from the accumulation layer into the QW. For the sake of simplicity, we consider the potential shown in fig. 1, which has standard solutions [10]. QW1 represents the emitter accumulation layer (typically with just one H H bound-state) and QW2 is the QW between the two barriers. The barrier thickness (b) of the samples studied [1] is 51 /~ and the barrier height (V0) is 550 meV (AlAs), so that we can treat the emitter accumulation layer and the QW as weakly coupled systems. To implement the formalism we consider the two separate subsystems (figs. lb and lc) and solve the Schr6dinger equation for each corresponding potential. If qt L and ~R are the multi-c'~mponent L and R wave-functions, we can obtain the tram, ition rate WLR by evaluating eq. (6) for z = 0 . Contrary to the case of electrons, there is no simple explicit expression for WLR for holes, and one has to resort to numerical calculations. The current Al,,(k II) due to energy and in-plane wavevector-conserving transtions from QW1 (L) into the nth state of QW2 (R) is A l , , ( k l l ) = e Wn(kII), where W,, is the transition rate. Combining this equation for A/,,(k I1) and eq. (3) gives Al,,(kll)-
2~-e h IM"(klI)I26(AE"(klI))'
where AE,, = E L - E ~ is the difference between the L and R energy levels. At a fixed voltage AV (see fig. la), the current I,,(AV) resulting from transitions into the nth state of QW2 is obtained by integrating Al,,(k II), using periodic boundary c(,:vlitions to determine the density of states in
E. C. Valadares et al. / Sequential tunneling of holes
411
dispersion curve for the emitter bound state,
E e(k II), and the dispersion curve for the nth !'--
HH
_
°
~V i !
0 (la)
R
:}W2
L
state of the QW. For a narrow QW, an applied bias (AV) basically shifts rigidly EL(k [I) tbr all k II, tuning the resonant tunneling, just as for electrons. Although the present considerations do not involve a magnetic field (B = 0), we can treat the effect of an in-plane field (B II y) as a perturbation and, if we take k v = 0, a block diagonalization of H e becomes possible [11]. The effect of B, in first order, is a relative shift in momentum space of the dispersion curves of QW1 and QW2 according to eq. (1) of [1], thus allowing the dispersion curves of QW2 to be scanned [1]. A detailed numerical analysis along these lines taking into account a more realistic potential for the emitter accumulation layer is in progress. Finally, we should mention some basic differences between the present formalism from its usual version for electrons. First, the transition rate WLR is ,+ "rv sensitive to mixing of HH and LH in the accumulation layer. It arises from the
(lb)
(lc) Fig. 1. (a) Approximate potential for the double-barrier structures studied in ref. [!]. (b) L subsystem (emitter accumulation layer). (e) R subsystem (QW between the two barriers).
k-space. In the axial approximation, this current is simply
2eLxL>. I,,(AV) =
h
O alu z
ul
~;
IM"(kll)lZ
k,, 1
× d A E , / d k II kll kl:" where k II = (k.~ + k~) !/2, LxL ~. is the cross-sectional area of the sample and d 2~E,/dh II is evaluated by considering the calculated dispersion curves of the bound-states of QW1 and QW2 (see fig. 2). The values of k II ~ (0 < k II ~ < k F ) which are solutions of the energy conservation equation ~ X E , , ( k l l i ) = 0 , can be interpreted graphically by looking for the intersections of the
k//(l) k~: ~/(A.U.) Fig. 2. In-plane dispersion curves for the emitter accumulation layer Et(k II)(chain curve) and OW states E~(k !1)(full curves), showing intersection point k (i) for a given AV.
412
E.C. Valadares et al. / Sequential tunneling of holes
fact that the transmission coefficient across the barrier is much bigger for LH, hence the major contribution of the LH component of the wavefunction to the tunneling current. Secondly, there is some difficulty to treat the 3D p-type electrodes because in the bulk LH and HH are decoupled. In the case of a 3D electrode the implementation of the present formalism requires us to consider a subsystem consisting of an interface (a step potential), for which we cannot ascribe a well-defined state with a unique dispersion curve (a HH (LH) incident on an interface generates a reflected LH (HH)[12]). Nevertheless, the usual box approach used for electrons removes this difficulty in principle by providing an extra boundary condition which defines unique R states. The same approach used for 2D electrodes described above could then be extended to treat the 3D electrode as a limiting case. This work is supported by the SERC (UK). One of us (E.C.V.) greatly acknowledges the financial support of the Brazilian Research Council (CNPq).
References [1] R.K. Hayden, D.K. Maude, L. Eaves, E.C. Valadare~, M. Henini, F.W. Sheard, O.H. Hughes, J.C. Portal and L. Cury, Phys. Rev. Lett. 66 ~1oo I '/ 1749. [2] G. Bastard, in: Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Paris, 1988). [3] H. Schneider, H.T. Grahn, K.v. Kiitzing and K. Ploog, Phys. l¢'.ev. B 40 (1989) 10040. [4] R.K. Hayden, T. Takamasu, D.K. Maude, E.C. Valadares, L. Eaves, V. Ekenberg, N. Miura, M. Henini, J.C. Portal, G. Hill and M.A. Pate, Proc. 7th Intern. Conf. on Hot Carriers in Semiconductors (Hr.;IS-7), Nara, Japan, 1991, Semicond. Sci. Technol., to be published. [5] J. Bardeen, Phys. Rev. Lett. 6 (1961) 57. [6] D.A Broido and L.J. Sham, Phys. Rev. B 31 (1985) 888. [7] C.Y.P. Chao and S.L. Chuang, Phys. Rev. B 43 (1991) 7027. [8] E. Merzbacher, in: Quantum Mechanics, 2nd ed. (Wiley, New York, 1970). [9] R. Wessel and M. Altarelli, Phys. Rev. B 39 (1989) 12802. [10] L.C. Andreani, A. Pasquarello and F. Bassani, Phys. Rev. B 36 (1987) 5887. [11] G. Piatero and M. Altarelli, Phys. Rev. B 39 (1989) 3758. [12] S.L. Chuang, Phys. Rev, B 40 (1989) 10379.
surface science
Sequential tunneling of holes in p-type semiconductor double-barrier heterostructures E.C. Valadares, F.W. S h e a r d and L. Eaves Department of Physics, Unicersityof Nottingham, Nottingham, NG7 2RD, UK Received 7 June 1991" accepted for publication 31 July 1991
We propose an extension of Bardeen's transfer Hamiltonian formalism to treat the sequential tunneling of holes in p-type semiconductor heterostructures displaying weakly coupled QW's. Our discussion is aimed at modeling recent resonant magnetotunneling experiments on double-barrier heterostructures displaying an emitter accumulation layer and wide barriers.
Recent resonant magnetotunneling spectroscopy (RMTS) experiments [1] have revealed important features of the valence band of p-type quantum-well heterostructures, such as the anticrossing and negative mass behaviour which arises from the spin-orbit term in the Luttinger Hamiltonian [2]. These and previous transport and optical experiments on G a A s / A I A s superlattices [3,4] strongly support the sequential tunneling of holes in such heterostructures. A formalism incorporating the sequential character of tunneling is thus of great interest for the discussion of such experiments. Traditionally, Bardeen's transfer Hamiltonian formalism has been used to model sequential tunneling of electrons (one-band systems) in n-type semiconductor heteros,ructures [5]. In the present paper, we outline an extension of this formalism to treat tunneling of holes in multiband systems within the effective-mass approximation. In order to discuss Bardeen's formalism in the context of holes, we consider the four-band Luttinger effective-mass approximation and neglect the spin-orbit split-off band. A further simplification which can be introduced is the axial (or cylindrical) approximation [6]. Under this approximation, the 4 × 4 Luttinger Hamiltonian can be diagonalized into 2 × 2 blocks by means of a suitable unitary transformation [6]. The formal-
ism can treat the complete Luttinger Hamiltonian, but these simplifications save considerable computing time without losing the main feature of band mixing between heavy holes (HH) and light holes (LH). Nevertheless, the split-off band (A = 340 meV) has to be included if states with comparable energies are considered. The Luttinger Hamiltonian can then be written [2], HL(k ) = a k 2 + /3( k . L) 2 + V( z),
(1)
where L = ( L , , L v,L_) are the 4 × 4 angular 3 momentum matrices for quantum number L and V(z) is the heterostructure potential. As discussed previously [7], one can define a current density operator J related to the Luttinger Hamiltonian. For our purposes, only the z-col.~ponent of J, J~, is relevant. It is used to obtain the solutions of the Schr6dinger equation for H L, which involve the matching of the wave-function and the flux at each interface [2]. The matrix expression for J. associated with H L reads [7] _
1
(J:)ij,= ~(D~. ~, + D~.)k~,
(2)
where D is the inverse effective-mass tensor asso3 ciated with eq. (1) and j,j' refer to the spin basis states (3, ~), /3 ~ {x, y, z} and _ 2I' 5~ , . the sum over repeated indices is implied. This operator is required to calculate tunneling cur-
0039-6028/92/$05.00 © 1992 - Elsevier Science Publishers B.V. and Yamada Science Foundation. All rights reserved
410
E.C Valadares et ai. / Sequential tunneling of holes
rents through Bardeen's formalism. The essential idea of this formalism is to treat tunneling in terms of transitions from left-hand (L) and righthand (R) states. The transition rate WLR between subsystems L and R then reads 2~" WLR =
7// I MLR 126(E L - ER) ,
(3)
where (4)
MLR = ihJLR~kllt,kilR ,
and k II refers to the transverse components of the wave-vector in the R and L subsystems. For electrons, the expression for /LR is
-ih (d~L ~ ~ * -
JLR= 2m 8
dz
R
~L
dqt~) dz :=, ,
(5)
where m B is the effective mass in a region (barrier) common to the R and L subsystems in which expression (5) is evaluated ( z = s ) . This expression has the same form as the probability flux carried by a single wave-function (see, e.g., ref. [8]). From expressions (3) and (4) it is clear that conservation of energy and in-plane momentum, k II = (kx, k:), during tunneling is required. The corresponding expression for JLR for the Luttinger Hamiltonian, which involves multibands, is obtained in the same way as for electrons, since Bardeen's formalism essentially corresponds to a first-order timz-dependent perturbation theory. It reads 1
JLR = -- ~ih(qt~Jflt L - q t L J : ~ ) z= ~,
(6)
where now qt R and qr L refer to four-component (or six-component, if the split-off band is included) wave-functions, which are solutions of the Schr6dinger equation for subsystems R and L, respectively. Although the application of the formalism to holes invoives essentially the same steps used for electrons, there are some important differences which arise from the mixing between heavy- and light-hole states. For systems involving the transfer of holes from one quantum well to another, the application of the formalism is quite similar to its electronic counterpart. Time-dependent models of tunneling of electrons were primarily developed
to calculate the transition rates between quasibound L and R eigenstates, such as 2DEG formed in the emitter accumulation layer of n-type single- and double-barrier heterostructures and states in the QW. Such states carry no current in the direction of confinement and, therefore, cannot be treated using a time-independent formalism [9]. We address our discussion to the p-type GaAs/A1As double-barrier structures studied recently by RMTS [1]. They comprise a 2DHG in the emitter accumulation layer and QW states which are probed by the RMTS. Since these experimei3ts provide the dispersion curves of the QW, we expect the tunneling current to be primarily determined by the transfer of charge from the accumulation layer into the QW. For the sake of simplicity, we consider the potential shown in fig. 1, which has standard solutions [10]. QW1 represents the emitter accumulation layer (typically with just one H H bound-state) and QW2 is the QW between the two barriers. The barrier thickness (b) of the samples studied [1] is 51 /~ and the barrier height (V0) is 550 meV (AlAs), so that we can treat the emitter accumulation layer and the QW as weakly coupled systems. To implement the formalism we consider the two separate subsystems (figs. lb and lc) and solve the Schr6dinger equation for each corresponding potential. If qt L and ~R are the multi-c'~mponent L and R wave-functions, we can obtain the tram, ition rate WLR by evaluating eq. (6) for z = 0 . Contrary to the case of electrons, there is no simple explicit expression for WLR for holes, and one has to resort to numerical calculations. The current Al,,(k II) due to energy and in-plane wavevector-conserving transtions from QW1 (L) into the nth state of QW2 (R) is A l , , ( k l l ) = e Wn(kII), where W,, is the transition rate. Combining this equation for A/,,(k I1) and eq. (3) gives Al,,(kll)-
2~-e h IM"(klI)I26(AE"(klI))'
where AE,, = E L - E ~ is the difference between the L and R energy levels. At a fixed voltage AV (see fig. la), the current I,,(AV) resulting from transitions into the nth state of QW2 is obtained by integrating Al,,(k II), using periodic boundary c(,:vlitions to determine the density of states in
E. C. Valadares et al. / Sequential tunneling of holes
411
dispersion curve for the emitter bound state,
E e(k II), and the dispersion curve for the nth !'--
HH
_
°
~V i !
0 (la)
R
:}W2
L
state of the QW. For a narrow QW, an applied bias (AV) basically shifts rigidly EL(k [I) tbr all k II, tuning the resonant tunneling, just as for electrons. Although the present considerations do not involve a magnetic field (B = 0), we can treat the effect of an in-plane field (B II y) as a perturbation and, if we take k v = 0, a block diagonalization of H e becomes possible [11]. The effect of B, in first order, is a relative shift in momentum space of the dispersion curves of QW1 and QW2 according to eq. (1) of [1], thus allowing the dispersion curves of QW2 to be scanned [1]. A detailed numerical analysis along these lines taking into account a more realistic potential for the emitter accumulation layer is in progress. Finally, we should mention some basic differences between the present formalism from its usual version for electrons. First, the transition rate WLR is ,+ "rv sensitive to mixing of HH and LH in the accumulation layer. It arises from the
(lb)
(lc) Fig. 1. (a) Approximate potential for the double-barrier structures studied in ref. [!]. (b) L subsystem (emitter accumulation layer). (e) R subsystem (QW between the two barriers).
k-space. In the axial approximation, this current is simply
2eLxL>. I,,(AV) =
h
O alu z
ul
~;
IM"(kll)lZ
k,, 1
× d A E , / d k II kll kl:" where k II = (k.~ + k~) !/2, LxL ~. is the cross-sectional area of the sample and d 2~E,/dh II is evaluated by considering the calculated dispersion curves of the bound-states of QW1 and QW2 (see fig. 2). The values of k II ~ (0 < k II ~ < k F ) which are solutions of the energy conservation equation ~ X E , , ( k l l i ) = 0 , can be interpreted graphically by looking for the intersections of the
k//(l) k~: ~/(A.U.) Fig. 2. In-plane dispersion curves for the emitter accumulation layer Et(k II)(chain curve) and OW states E~(k !1)(full curves), showing intersection point k (i) for a given AV.
412
E.C. Valadares et al. / Sequential tunneling of holes
fact that the transmission coefficient across the barrier is much bigger for LH, hence the major contribution of the LH component of the wavefunction to the tunneling current. Secondly, there is some difficulty to treat the 3D p-type electrodes because in the bulk LH and HH are decoupled. In the case of a 3D electrode the implementation of the present formalism requires us to consider a subsystem consisting of an interface (a step potential), for which we cannot ascribe a well-defined state with a unique dispersion curve (a HH (LH) incident on an interface generates a reflected LH (HH)[12]). Nevertheless, the usual box approach used for electrons removes this difficulty in principle by providing an extra boundary condition which defines unique R states. The same approach used for 2D electrodes described above could then be extended to treat the 3D electrode as a limiting case. This work is supported by the SERC (UK). One of us (E.C.V.) greatly acknowledges the financial support of the Brazilian Research Council (CNPq).
References [1] R.K. Hayden, D.K. Maude, L. Eaves, E.C. Valadare~, M. Henini, F.W. Sheard, O.H. Hughes, J.C. Portal and L. Cury, Phys. Rev. Lett. 66 ~1oo I '/ 1749. [2] G. Bastard, in: Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Paris, 1988). [3] H. Schneider, H.T. Grahn, K.v. Kiitzing and K. Ploog, Phys. l¢'.ev. B 40 (1989) 10040. [4] R.K. Hayden, T. Takamasu, D.K. Maude, E.C. Valadares, L. Eaves, V. Ekenberg, N. Miura, M. Henini, J.C. Portal, G. Hill and M.A. Pate, Proc. 7th Intern. Conf. on Hot Carriers in Semiconductors (Hr.;IS-7), Nara, Japan, 1991, Semicond. Sci. Technol., to be published. [5] J. Bardeen, Phys. Rev. Lett. 6 (1961) 57. [6] D.A Broido and L.J. Sham, Phys. Rev. B 31 (1985) 888. [7] C.Y.P. Chao and S.L. Chuang, Phys. Rev. B 43 (1991) 7027. [8] E. Merzbacher, in: Quantum Mechanics, 2nd ed. (Wiley, New York, 1970). [9] R. Wessel and M. Altarelli, Phys. Rev. B 39 (1989) 12802. [10] L.C. Andreani, A. Pasquarello and F. Bassani, Phys. Rev. B 36 (1987) 5887. [11] G. Piatero and M. Altarelli, Phys. Rev. B 39 (1989) 3758. [12] S.L. Chuang, Phys. Rev, B 40 (1989) 10379.