Solid State Communications, Vol. 88, No. 4, pp. 309315, 1993. Printed in Great Britain.
00381098/93 $6.00 + .00 Pergamon Press Ltd
HARTREE ENERGY IN SEMICONDUCTOR QUANTUM WELLS M. Combescot, O. BetbederMatibet, C. Benoit fi la Guillaume and K. Boujdaria G.P.S., Universit6 Denis Diderot et Universit6 Pierre et Marie Curie, CNRS URA 17, Tour 23, 2 place Jussieu, 75251 Parix Cedex 05 (Received 6 June 1993; accepted for publication 12 July 1993 by G. Bastard) We calculate the Hartree energy of an electron gas in a doped semiconductor quantum well. We compare its exact value with its perturbative expansion. We show that for densities low enough to have one subband filled only, the exact calculation is unnecessary, the perturbative expansion with one or possibly two terms being already quite good. 1. INTRODUCTION THE VARIOUS ways [13] to calculate the Hartree energy can appear quite different at first. In all of them, however, the Hartree energy is finally related to the part of the Coulomb interaction which depends on spatial charge distribution. In systems with local neutrality such as the well known 3D electron gas in which the electron negative charges are compensated everywhere by a positive jellium, the Hartree energy reduces to zero, the jelliumjellium and electronjellium interactions cancelling exactly the q = 0 diverging term of the electronelectron interaction. In systems with global neutrality only, a finite contribution remains, which is related to local nonneutrality. One example of such locally nonneutral systems is the quasi2D electron gas of doped semiconductor quantum wells: the electrons and the ions are either located in separate layers or, when they are in the same layer, a positive charge excess does exist at the layer surface since the ion density is usually constant in the well while the electron wave function has to adjust itself to the potential barrier. In this paper, we calculate the Hartree energy of quasi2D electron gas in semiconductor quantum wells, as a function of the electron density, ns = N / S , and well width, a, using two different approaches: (i) In a perturbative approach [2], we separate the full ionion, ionelectron and electronelectron Coulomb interaction into excitations at zero and nonzero momentum transfers. The first ones, corresponding to intrasubband and intersubband transitions at constant k, depend on spatial charge delocalization. The energy associated to them is the Hartree energy. We calculate it using perturbation theory.
(ii) In a variational approach [1], we minimize the average value of the full Hamiltonian, calculated with a trial Nparticle wave function taken as a simple product of oneparticle wave functions (i.e. without any antisymmetrization procedure). The oneparticle wave function corresponding to the minimum verifies an integrodifferential equation which can be intuitively viewed as a Schr6dinger equation for an electron in the average potential [3] induced by the ions and the other electrons. This integrodifferential equation can be solved perturbatively: the resulting Hamiltonian average value is then exactly the Hartree energy obtained through the first approach. This integrodifferential equation can also be solved numerically: Comparison between the two results allows to establish the range of validity of the perturbative approach. As the Hartree energy is physically related to spatial charge delocalization, we expect the pcrturbatire expansion to be good for small electron density and well width. More precisely, we show that the dimensionless parameter associated to the Hartrec energy expansion is K2a 3 
7r4ao
(1)
where K = (2~rns)U2 is the 2D Fermi momentum and a0 = h2/me 2 is the Bohr radius. As physically expected, this parameter is proportional to the electron density ns and to the Coulomb strength e2 ,~ a~ I , the a 3 factor insuring AH to be dimensionless. The additionnal ~r4 factor, which will be shown to appear systematically in the perturbation expansion, is useful to keep AH small even for rather dense systems. In quantum wells with the n = 1 subband filled
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H A R T R E E E N E R G Y IN S E M I C O N D U C T O R Q U A N T U M WELLS
only, the electron density is limited by the condition:

h2(Tr 2 ) h247r 2 2m a ~ + K 2 <2m 'T=~Ka
 ~re2N2
(2)
This corresponds to Au < 3/7r 2 ,,~ 0.3 if a/ao < 1. We will show that for such Au's a good quantitative agreement with the numerical nonperturbative result is obtained with the two lowest order terms of the Hartree energy expansion only (the first order term being already enough when the ion distribution is symmetric with respect to the electron layer). This physically means that, for densities such that condition (2) is fulfilled, the wave function deformation induced by the Hartree potential to restore local neutrality stays small. As the Hartree energy depends on local nonneutrality, it depends on the ion localization. Indeed, the precise ion distribution is important when electrons and ions are in the same layer. However, when electrons and ions are located in separated layers, the Hartree energy (beside a trivial first order term) will be shown to only depend on the difference between the numbers of ions located on the left and on the right hand side of the electon layer, and not on the electronion distance. (This basically comes from the fact that the electrostatic field of a plane charge distribution does not depends on the distance to the plane.) Consequently, we will calculate the Hartree energy in the two main cases:  when ions are uniformly distributed inside the electron layer;  when ions are located outside the electron layer, with any kind of distribution.
Vol.
88, No.
4
f f dz, dz2P(Zl)P(Z2)e'7(z'z') s
JJ
S
n
[~JJdzldZ2lZ, z2[p(zl)p(z2)].
(4)
The ionelectron and electronelectron interactions can be written in second quantization, using creation operators a ~ associated to the free electron wave function in the well. For infinite potential barrier, these are given by
the corresponding energies being
h2[
~kq£n=~m
k 2 t   ~   ] .
(6)
The electronion interaction then reads N
Vei = j~__l J d3r'n(rt) V(rj  r' )
S"
T.n k ,nlk , ,a+ , , I ntkla I a nlkltrj
(7)
nn' kl k't el
with
Tn, k,;n',k'
=
Id3rl(I)*,k,(rl) x [Id3r2n(r2)V(rlr2)]On,,k,,(rl)
(8)
27re2N [ 1
2. PERTURBATIVE APPROACH
 I J dz, dz2lz,  zzl~o*.,(z,)~on,(zt)p(z2)]
2.1. Hartree interaction In order to properly eliminate the divergences arising from the long range character of the Coulomb interaction, we consider a globally neutral system made of N electrons and N ions, and we replace the true Coulomb potential e2/r by a screened one [2] V(r) =
e2erlr r
e2 [
eiq't 772
= 27r2 j d3q q2 +
'II
d3rl d3r2n(rl)n(r2) V (r I  r2)
1
Vee=~ ~
V(rjrj,)
(9)
./~j~:t=l
(3)
r/ being a vanishingly small positive constant. This trick allows to follow the divergences and verify that they do cancel. Using equation (3), we can write the ionion interaction, for an ion density n(r) = nsp(z), as Vii = ~
When written along the same procedure, the electronelectron interaction
can be divided into two parts which correspond respectively to zero and nonzero q momentum transfers. For vanishingly small r/, the q = 0 part reads
:
:E nl~n2nt2kl k2trto'2 +
+
X Unl~ ;n2n,2an,k,a,an2k2a2an,2k2a2an;k,a,
(10)
H A R T R E E E N E R G Y IN S E M I C O N D U C T O R Q U A N T U M WELLS
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with
defined as
27re2[~6.,.,,.6,,2.,2 U., .,, ;.2.; = g
v =  I 1 dzl dz2 lz'  z2l [~(zt) 
IIdZl
The q # 0 part, which stays finite when ~/~ 0, gives rise to the exchange and correlation energies and will not be considered here. From equations (4, 78, 1011), we can check that the divergent 1#1 terms of the full Coulomb interaction, VcouI = Vi+ Vei+ V~e, cancel since the electron number, N, can be replaced by the operator
[4] (]2)
ankaanka. +
nka
The remaining finite part of ["coul / ( q = 0 ) ' which corresponds to the Hartree interaction, reads:
Vc~qu?O) 7re2a :
s
(15 /
X [/ge(Z2)  p ( 2 2 ) ]
dz21z,z21qo*.,(z,)~o.;(z])~o'.~(zz)~.~(z2)].
where l%(z) is the average electron density
(ll)
~r = Z
311
Z S_,
?/IF/t ?/2Plt9 klk2 
+
o 1 (7 2
+
X an,klcrlan2k:2an,2k:a2an,tklal (13) where the dimensionless l'V~:,;n:.~ coefficient is given by
De(z) =
l/a
forO
0
otherwise
(16)
and to set
I.V~,.,,;n2~ = v6n,.,, 6n2n'2 + w.,.., ;"24"
(17)
The v term depends on the precise ion distribution. It vanishes when the ions are uniformly distributed in the electron layer while it becomes quite large when the ions are far away. Due to equations (12). this term induces a constant (#[N2rre2a/S]) contribution to the Hamiltonian. which can be seen as a mere electrostatic shift which takes most of the charge delocalization into account. We can forget about it. provided that we add this contribution to the energy. The remaining Wnln,,;n2,n ~ term is explicitly calculated in the appendix. When the ions are outside the electron layer, it only depends on the ( N L  NR) difference between the numbers of ions located at the left and at the righthand side of the electrons. From it, comes the energy change induced by the electron wave function adjusting to the ion and electron field. 2.2. Hartree energy expansion
=II z, X [~0" 1 ( Z l ) ~ n ; ( g l )
a 
From perturbation theory, we easily get the two lowest order terms of the Hartree energy expansion: fl(Zl)6n,n; ]
x [qo~2(Zz)q0n~(z2)  p(z2)6n2n,2]
(14)
vc(q=O) describes intrasubband (n'l = hi) and intersuboul band (n'l # nl) transitions at constant k, i.e. without any q = (qx, qy) momentum transfer. We can, however, note that, while An # 0 transitions would merely correspond to qz = (TrAn)/a # 0 excitations in a large well (3D) limit, the existence of intrasubband excitations are specific to the Hartree interaction. Their weight for a n subband depends on [q~,(z)]2p(z)], see equation (14), so that the transitions inside the n subband are physically related to the local charge excess which would exist if these electrons were all occupying the same free nth electron wave function. We will see that these intrasubband transitions play a key role in the Hartree energy. In the following, it will appear appropriate to extract from W.,.,;.2.~ a constant diagonal term
E ( l ) = (0[Vc(oq~10)[0) = N(Nrre2a~(v+Wll; 11)
\s:
Eft ) = ~
I(mlVc°ul . . . .[0)l
m~O Eo Era
(18)
N NTre a
41Wll;l,] 2 __
~
'I  en (19)
where ]0) is the unperturbed ground state corresponding to N electrons in the lowest n  1 subband. The Hartree energy can be seen as a change of the el localization energy induced by nonlocal neutrality. From equations (18. 19). we get the first two terms of this shift expansion Ael = N  I E H = el [(Tr2v + ")'I)AH  ")'2A2 + . . . ] (20) ")'1 = 71"2WI1;11 2
72 = 4 Z n>l
wi,;,.I ¥i
I~
2
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H A R T R E E E N E R G Y IN S E M I C O N D U C T O R Q U A N T U M WELLS
n
where An is the dimensionless parameter defined in equation (1): Note that, as each Wntn,t;n2n,2 matrix element contains a 7r2 prefactor (see the Appendix) and each energy denominator induces another 7r2 factor, a 7r4 factor has indeed to be included in the definition of the AH expansion parameter in order to get 7i prefactors of the order of 1. When the ions are uniformly distributed inside the electron layer, we get v = 0 and
7(in) 1 I =4
7~in)
=
~2
1 Ca)
(b)
11
n'
1
1
D
n
(21)
"
When the ions are outside the electron layer, we find _ (out) = 5 "h 4
,,[~out)
7r2
(
+
15(10"n2  81)  27r4 360
12(157r2)
NL NR
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(c) Fig. 1. First (a I. second (b) and third diagrams in v~qs1'~(wavy line). The (1) line an electron in the lowest n = 1 subband, (n) line represents an electron excited subband.
)2
(22)
~ 4.22 (NL NNR)2+0.20.
The v coefficient, which differs from zero in this case, depends on the precise ion localization. We find v = (a+b)/6a, when the ions are uniformly distributed in two identical adjacent layers (with b/2 width), and v = 2(a + b)/3a when they are uniformly distributed in one adjacent layer (with b width). We do check that besides the 7r2v first order contribution which can possibly be much larger than that one (if the ions are far from the electrons), the other prefactors of the AH expansion are of the order of one, so that equation (20) should give a result close to the exact one for densities low enough to have the n   1 subband filled only (i.e. AH < 0.3a/ao). We even see that the quadratic term is negligible for such densities, except when the ions are all on one side of the electrons. The different terms of the Hartree energy An expansion can also be calculated with the use of Green's function formalism and Feymann graphs [2]. The first and second order terms correspond to the diagrams of Figs l(a) and (b). The two extreme balloons represent unperturbed electrons in the n = 1 subband, while the inbetween bubble corresponds to the excitation of an electron from the n  1 to a n # 1 subband. The three types of third order terms are shown on Fig l(c). The summation of these Hartree diagrams can be performed to all orders in AH by
order (c) represents while the in n :A 1
using a selfconsistent [2] procedure. It is, however, simpler and physically more transparent to derive the resulting selfconsistent Hartree integrodifferential equation from a variational approach. 3. V A R I A T I O N A L A P P R O A C H We start with a H Hamiltonian composed of a kinetic part, a W(z) well potential and a full Coulomb interaction between electrons and ions. We look for the minimum of E¢(¢IHI~O), restricting to a normalized Nparticle wave function, ~, which is a simple product of normalized oneparticle wave functions Fk,(r)
Fki(r) = Z(z) [~seik~'P] .
(23)
For electrons in the lowest subband only, the Z(z) function is the same for all electrons, while the kl vectors cover all the different k values such that k < K. We calculate E¢ using the Coulomb potential given in equation (3). This again allows to discard divergent r/l terms in a clean way. We eventually find E~,_ 1 h2K 2 J N 2 2nT+ dz Z* (z)
h 2 d2 2m dz 2 +
]
W(z) Z(z)
[ IZ(z)12  p(z)] × [I/(z')l
2
p(z'
)1.
The functional minimization of E¢ relatively to
(24)
Z(z)
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H A R T R E E E N E R G Y IN S E M I C O N D U C T O R Q U A N T U M WELLS
(a)
0.07 0.08
~
( )1.2
0.06 
/
=. o.o5U j :¢
:: 0 .t 6
0.04 0.03 0.02
0.4 0.2

0.01 0
,,i
0
/
~
. . . .
0.05
i . . . .
0.1
i . . . .
0.15
r . . . .
0.2
t . . . .
0.25
0
~ . . . .
0.3
~'n
i . . . .
0
0.05
0.1
0.15
i . . . .
0.2
i . . . .
0.25
i . . . .
0,3
~'n
(C) 43=

~2 W
•
10
'' =' =
0
9, 9
0.05
. . . . . . . . . . . . . . . . . . . . .
0.1
0.15
0.2
0.25
0.3
~n
Fig. 2. Hartree energy in q unit as a function of the dimensionless parameter An defined in equation (1) (An is proportional to the electron density). Figures (a, b, c) correspond to three different ion configurations with respect to the electron layer, shown in the inserts. Results are presented for b = a. The solid lines correspond to the second order expansion given in equation (20, 21, 22) while the +'s correspond to the numerical resolution of equations (25, 27). The O's in Fig 2(c) correspond to the difference between the numerical result and the expansion (multiplied by 20). Excellent agreement is obtained for An < 0.3, which corresponds (for a well width equal to the Bohr radius) to electron densities such that the lowest n = 1 subband is filled only. leads to the following integrodifferential equation
eigenvalue as
A q = E'  q + lr2AHq I J dz dz' 'z  z'
h2 d 2 2m dz 2 + W(z)  27r2AHel
x [IZl(z)I 2 IZl(z' )l 2  p(z)p(z' )].
x J dz 'tzz'l (iZ(z,)12p(z'))]Z(z)=EZ(z) (25) since 7re2nsa =71"2AH(1, e I being the free electron localization energy in the n = 1 subband and An the parameter defined in equation (1). (The E Lagrange multiplicator arises from the normalization condition on Z(z).) Inserting the lowest solution, Z~ (z), of equation (25) into equation (24), we find that the Hartree energy defined as
A(I  (Eo)min N
(~ ~'I2K 2m 2 ~el )
(26)
can be written in terms of the corresponding E I
(27)
A corrective term to the E l  e 1 differences arises from the fact that the electronelectron and ionion interactions are counted twice in El. If An is small, we can solve equation (25) by perturbation. The lowest solution then reads Z I (z) = ~oI (z) + Z ( a n A n +/3,A~ +...)~o,(z).
(28)
n
It is easy to cheek that the resulting An expansion of the Ael Hartree energy is identical to the one obtained from the perturbative approach, equation (20). Equation (25) can also be solved numerically for any AN. The resulting Ael is shown on Figs 2(a), (b) and (c), as a function of An, for three particular ion configurations: the ion density is constant inside the electron layer (a), in two adjacent layers (b), or in one adjacent layer (c).
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H A R T R E E E N E R G Y IN S E M I C O N D U C T O R Q U A N T U M WELLS
On the same figure, is aho plotted Ael as obtained by the perturbative expansion up to second order in An, (equation (20)). We see that for densities low enough to have the n = 1 subband filled only (i.e., for An < 0.3 if a/ao = 1), the quantitative agreement is very accurate, so that the nonperturbative calculation is unnecessary. Moreover, the curves being essentially straight lines, the linear term of the AH expansion is very close to the exact result in most cases. We however see a small deviation from linearity when the ions are all on one side of the electron layer (i.e. when the local charge nonneutrality is the largest). In this case, the contribution of the quadratic term cannot be neglected at the largest densities. However, when this term is included, the perturbative result is very close to the selfconsistent one. There only remains a very small (cubic) discrepancy as explicitly shown on fig. 2(c), which could possibly be taken into account by including the )~3 term of the perturbative expansion. 4. C O N C L U S I O N
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APPENDIX
Calculation of W,,,e:: 6 From equations (1416), we can rewrite wn,n,:n2n,2 as
WnLn,t;n2n'2 = tn,6 6n2/i~ + t/i26.6n,n(+u.,5;.2.,2
(A1)
with
× [~oT(Zl)~O.;(z~)  ~(z~)& ~, lp(z2)
(A2)
u/i'/i'L;"2"~ =  I I dzl dz2 lzl 
× [~.; (zt):.; (zt)~o.~(z2)~/i~(z2)  Pc( zl )Pe(Z2)~n,n'l ~/i2/i'2]
(A3)
where De(Z) is the average electron density given by equation (17), and qa/i(z) is the unperturbed wave function defined in equation (5). t/i/i, depends on the precise ion localization, while u/i,~,::~ does not.
(i) When the ions are uniformly distributed We have calculated in two different ways, the inside the electron layer; p(z) = De(z), and tnn' is given Hartree energy of a quasi2D electron gas in a doped by semiconductor quantum well. In the first way, we truncate the Coulomb interaction to keep only t(in) l + (1)n+/i' [ . ( 1 1 ] #n') intrasubband and intersubband transitions at zero ~in'  ~ n  n') 2 ( n + n') 2 (n momentum transfer; the Hartree energy is obtained by calculating perturbatively the energy corrections ton) 1 (A4) 2n27r2. induced by these q = 0 transitions up to second order. In a second way, we keep the full Coulomb We note that ,(in) ~ n n t couples bands of same parity only. interaction and calculate the minimum of the (ii) When the ions are outside the electron layer, Hamiltonian expectation value, in a variational t/in , is given by way, using "truncated" wave functions, (i.e. functions which are simple products of oneparticle wave t(°ut) (  1 ) n + d  1 [ ! 1 1 /i/i' 7r2 (n n')2 (n + n')2 functions and thus do not follow Pauli exclusion). These oneparticle trial functions are found to verify an integrodifferential equation. From the numerical X (NLNNR) (n#n') resolution of this equation, we also deduce the Hartree energy. t(out) = 0 (A5) /in By comparing the results obtained with these two (out) now couples bands with different parities only. nVI t methods, we show that, for density low enough to (iii) Turning to u,l,,;/i2~, it is easy to show that have the lowest subband filled only, the numerical selfconsistent approach is unnecessary. In most cases, a good quantitative value of the Hartree u/i,/i,I;/i:~ = ~ [I/i~_/i, ,/i~_/i~ + I~, +e,,/i~+.~ energy is already obtained from the first order term  In,/i,, ,/i:+/i~  I/i, +/i, :,2/i~] (A6) of the perturbative approach. However, when the ions are all on one side of the electron layer, and with when the electron density is close to the value In*m, = tSIml,Ira'I/m2 (m, m' ~ O) corresponding to the filling of the second subband, the second order term of the perturbative expansion, Ion* = [1 + (l)n*]/rn 2 (m # 0); (A7) has to be included in order to get a very accurate I0o=0 result for the Hartree energy. nn
~
Vol. 88, No. 4
H A R T R E E E N E R G Y IN S E M I C O N D U C T O R Q U A N T U M WELLS REFERENCES
1. 2.
N. Ashcroft & D. Mermin Solid State Physics p.330, Saunders Company ,(1976). A. Fetter & J. Walecka, Quantum Theory of Manyparticle Systems. p.121, McGrawHill (1971).
3. 4.
315
G. Bastard, Wave Mechanics Applied to Semiconductor Heterojunctions. p. 161, Editions de Physique (1990). For any Nparticle state I~), we have ~r 1~) = + + a N Ik~) and ~n,k]~, an, k,atan2k2a2a~k2a2 ~k,a, 1~) n2k2cr2
N(N
N21