- Email: [email protected]
The Landau level density of states as a function of Fermi energy in the two dimensional electron gas
~ )
Solid State Communications, Vol. 81, No. 10, pp. 821-825, 1992. Printed in Great Britain.
0038-1098/9255.00+.00 Pergamon Press plc
THE LANDAU LEVEL DENSITY OF STATES AS A FUNCTION OF FERMI ENERGY IN THE TWO DIMENSIONAL ELECTRON GAS R.C. Ashoori* and R.H. Silsbee
Laboratory of Atomic and
Solid State Physics, Cornell University, Ithaca, New York 14853, USA
(Received 11 December 1991 by A. Pinczuk) We have made quantitative determinations of the density of states (DOS) of Landau levels in a 2d system whose density can be varied using a gate bias. A novel technique employs two normalization conditions to extract the DOS as a function of Fermi energy in the 2d electron gas from capacitance data. No knowledge of sample parameters is required except for the sample area. In contrast with previous results, we find that Lorentzian lineshapes give excellent fits to the Landau level structure without need for added background and with llnewidths that are independent of magnetic field strength.
Since the early magnetoconductance work of Fowler et. al.1 in silicon MOSFETs, and especially after the discovery of the quantum Hall effect, there has been a great effort to understand the Landau level density of states (DOS) in two dimensional (2d) systems. 2 A clear picture of the Landau level DOS should be an aid for understanding features in the quantum Hall effect. The existence of an interlevel density of localized states is thought to be of critical importance for the occurrence of Hall plateaus, s There have been several measurements of the Landau level DOS with experiments using specific heat, 4,5 magnetization,B and capacitance. 7-9 Although a clear picture of the Landau level DOS has not emerged, several of these experiments suggest that the Landau level DOS can be described by a Gaussian shape (sometimes with an added ad hoc constant background) whose width depends on magnetic field. Previous experimental results 6 have indicated a B 1/2 dependence of the width of lines in agreement with the short range scattering model of Ando and Uemura. ~° Theoretical arguments have predicted elliptical lineshapes, 1° Gaussian lineshapes, 13 or lines whose shapes depend in a complicated way on the structure of the sample in which the 2d electron gas exists. 12 This Solid State Communication reports a series of capacitance measurements which give significantly different results. Specifically, Lorentzian lineshapes give excellent fits to the Landau level DOS in three samples with different doping profiles, and the widths of these lines are independent of magnetic field. We have developed a new technique for determination of the Landau level DOS. The method allows transformation of capacitance data directly into a quantitative measurement of the density of states without need
for any sample parameters (or even the dielectric constant of the medium) except for the lateral area of the sample which is well known. Even the effects of band bending in the top and bottom gates of the sample are automatically included in the analysis. Previous capacitance determinations of the Landau level DOS have been susceptible to large errors because of uncertainties in the values of sample parameters, as noted by Smith et. al. 13 Further, our experiments are done in samples in which the 2d electronic density can be varied by means of a gate bias. Our analysis explicitly yields the Landau level DOS as a f u n c t i o n of Fermi energy in the 2d electron gas. This allows immediate comparison of the DOS results discussed here with models that describe the DOS as a function of energy in the 2d gas, in contrast with other experiments which typically measure the DOS at the Fermi energy for a constant 2d electron density as the magnetic field strength is swept. Fig. 1 shows a conduction band edge diagram for one of the three samples used in the present study and is typical of all the samples. The samples are grown on n + GaAs conducting substrates using molecular beam epitaxy. They are of the same type that we have used in tunneling experiments. 14 Grown on top of the substrate are a GaAs undoped spacer layer, an A1GaAs tunnel barrier, a 150,~ wide GaAs quantum well, a thick AIGaAs "blocking barrier", and finally a heavily n doped GaAs top contact layer (gate). In some samples the blocking barrier, which allows no electrical conduction, contains a meg'ion of Si doping. Ohmic contact is made to the gate and the substrate. Table I lists various growth parameters for the samples used. The capacitance of these devices is a function of frequency. In the simplest picture, only valid for infinite DOS in the well, charge transfer between the quantum well and the substrate can be described in terms of a parallel R C circuit, with R the tunneling resistance and
* Present address: AT&T Bell Laboratories, Murray Hill, N.I 07974. 821
822
LANDAU LEVEL DENSITY OF STATES
Vol. 81, No. 10
Table 1. Growth parameters for the saxnples.
A
30
160
1550
Si dopants (from well edge) (/~) 100-200
B
30
133
800
no dopants
no dopants
C
150
150
800
150-500
6x1017
Sample
GaAs Spacer layer (~)
Tunnel Blockin~ Barrier (/~) Barrier (A)
C the capacitance between the 2d electron gas and the substrate. At frequencies small compared to 1/27rRC there is sufficient time for equilibrating charge to traverse the tunnel barrier during one half cycle of the measuring voltage, while at high frequencies little charge is transferred. We refer to the low and high frequency limiting values of the capacitance as CIo~ and Chigh respectively. In our experiments, the device capacitance is measured as a function of frequency over the range from 15 Hz to 30 kHz. This is done for various values of the gate bias, magnetic field applied perpendicular to the plane of electrons in the well, and temperature. Fits, based on a model described elsewhere] sAs are made to the capacitance to extract Crow and Chig~. The range of frequencies over which we measure the capacitance is sufficiently wide that at the lowest and highest frequencies the capacitance has essentially obtained its limiting values. Cto~ depends on the DOS in the quantum well, but Chigh does not. If the DOS in the well were infinite, G'tow would simply be determined by the sample area, the dielectric constants, and the spacing between the 2d plane and the gate. In contrast, if the DOS were equal to zero, there could be no charge transfer between the substrate and the well, and Crow would be equivalent
to Chigh. As the density of states in the well is varied between zero and infinity, Ctoto varies monotonically and predictably from a lower limit at zero DOS equal to Chigh to an upper limit for infinite DOS. Fig. 2 shows Clow and Chiah for a sample with no doping in the blocking barrier as a function of gate bias on the device. The results axe for 2 T applied magnetic field and a temperature of 2.l K. In the region below -80 mV in the figure, the quantum well is depleted of electrons and the DOS in the well is zero. In this case C~o,,, and Chigh have the same value. As the gate bias is increased, electrons begin to enter the well (the bound state energy in the well drops below the Fermi energy in the substrate), the DOS increases from zero, and Cto,o increases sharply. As the gate bias is increased further, Cto~, begins to oscillate as a function of gate bias. These oscillations arise due to the Landau level DOS in the well, with peaks at the Landau level centers and minima between levels. Ctow oscillates about its zero field value. Notice that Chigh contains no oscillations, as expected since Chigh is insensitive to the DOS in the well. Two equations provide interpretation of these results. T M One gives the "lever-arm" which relates the applied gate voltage V~t~ to the bound (ground) state energy in the well Ubou,,d measured relative to the fixed
"l Blocking Barrier
Gale
Well
Tunnel
Dopant Concentration (cm-3) 5x1017
'
'
'1
'
Subslrate
Barrier
1 I 1' 1
x x ×
C" v~
x x
0 E 0
x x x
X"
I
x x
"/".................. L ~
x × x × ×
. . .. .. .. .. .. ..
,~t'#
-100
0
100
i
i
200
300
t
400
500
600
Gate Voltoge (rnV)
Figure 1. The essential structure of the samples. Electron transfer through the tunnel barrier brings the electron gases in the substrate and the quantum well into equilibrium. The density of electrons in the quantum well may be varied through the application of a gate bias (Vg~t, in the figure).
Figure 2. Capacitances Crow and Chigh for sample B at 2.0 T and 2.1 K. The two dashed vertical lines are limits of integration for the normalization condition which :letermines Cg. . . . We integrate over two Landau levels :ather than one for higher precision.
Vol. 81, No. 10
LANDAU LEVEL DENSITY OF STATES
Fermi energy in the substrata (which, because of equilibration through the tunnel barrier, is the same as the Fermi energy in the quantum well). The lever-arm is given by
dUbo~,,~d d(eVga,e)
Crow -
C,eo,.,
C,o~ -1)(1 (C--~igh
rlCgeom). (1) Ae 2
Here e is the magnitude of the electron charge. The other equation is for the density of states g which is obtained through • Clow Ae=g = Cgeom( ~
_ 1) d(eVg~,,,)
dUbo~,~d"
(2)
A is the area of the sample which is known to within 1%. These equations involve two model parameters. Cg~om is the capacitance of the quantum well to the substrata in a model in which the DOS is infinite and the well charge is taken to lie at the center of charge of the real charge distribution in the well. 7/is a parameter describing the variation in bound state energy of the well associated with the details of the distribution of the electron density in the well as it is filled. 0 requires a brief explanation. Our analysis begins with a simplified model which considers electrons in the quantum well to lie in a sheet located at the mean position of charges in the well. The energy of the well bound state is initially taken to be a fixed energy E0 greater than conduction band edge energy U~, calculated in the sheet charge model, at this position, r] is the coefficient of a term qaw (where aw is the sheet electron number density in the well) which is subtracted from Uw + Eo to yield Ubo~,d, a more accurate result for the bound state energy. The r/a~0 term corrects the electrostatics in our analysis for the actual configuration of distributed charge in the well and also includes adjustments to the bound state energy for its quantum mechanical variation arising from changes in curvature of the band bottom of the well as the well is filled with electrons. We show elsewhere 15,16 that the rigidity of the shape of the electronic wavefunction in the well as the well is filled and the adequacy of first order perturbation theory in calculating changes to the bound state energy justify the inclusion of only this linear order correction. The exchange-correlation interaction9,17 also decreases E0 as the well is filled, and 7/absorbs the linear order variation in E0 as a function of density due to this effect. If reliable values of 77and Cg~o,, can be deduced from sample characterization and theoretical modeling then Eqs. 1 & 2 give the desired quantities. Unfortunately, modeling of the sample typically will no~ give sufficient accuracy in determination of these critical parameters to allow for confidence in the DOS results. Our method requires no such sample modeling. Two normalization conditions are available to determine C0.... and 7/ through knowledge of the experimentally measured quantities, Clow and Chig h. The degeneracy of a Landau level, when the sample is placed in magnetic
823
field perpendicular to the quantum well, is the number of flux quanta threading the sample per unit area times the spin degeneracy of two, or 2Be/h. Assuming a fixed value of Cg~om as the density in the quantum well changes, Eq. 2 may be integrated between adjacent minima of the measured quantity CIow/Chig h (corresponding to minima in the DOS) to determine Cg. . . . It is also known that the lever-arm, when integrated in Vg~,t~over a Landau level, must give tuzc/e, where wc is the cyclotron frequency. 7/is then determined by integrating Eq. 1 over gate voltage between adjacent Landau level peaks in the DOS. With Cg~om and r/ determined by these conditions, Eq. 2 gives the DOS directly from the data and knowledge of the sample area. With the inclusion of a small variation of Cg. . . . linear in gate bias and determined from examination of data taken in zero magnetic field, to account for the small changes with gate voltage of the values of 0 and Cocom,15,x6 this procedure provides a direct translation of the data in Fig. 2 to the Landau level DOS as a function of energy. The magnitude of this adjustment is in rough accord with computer simulations determining the variation in the mean position of electrons in the well, is calculations19 of the shift of the mean position of the substrata charge using the Thomas-Fermi approach of Baraff and Appelbaum, 2° and the expected variation in r/. We point out that the value of Cg~o,, extracted is in good agreement with expectations for the distance separation between the quantum well and substrata charge densities. In fields up to 4 T, the method
i
E~ 0 3 c~
o
-hOJc--
-9
-6
-3
-hwc-
0
-hwc-
3
6
Fermi Energy in Quantum Well (meV)
Figure 3. The circles are the DOS (On/O#) vs. Fermi energy in sample B at 2.0 T and 2.1 K, extracted using the analysis procedure described in the paper, from the data presented the data of Fig. 2. The vertical lines drawn are hcoc apart in energy. Note, one of the two unknown sample parameters in our model was adjusted so as to keep two of the peaks h~zc apart. The fact that other peaks also fall hwc apart affirms the consistency of our model. The solid curve is a fit to Lorentzian lineshapes. The fitting parameters are: Fj=1.47 rneV, F2=0.81 meV, F.~=0.74 meV, and P4=0.68 mcV.
824
LANDAU LEVEL DENSITY OF STATES
yields the same values of Cg~o,n independent of field strength. Using this procedure we plot in Fig. 3 the DOS based on the capacitance data of Fig. 2. We stress that the results of Fig. 3 have been obtained entirely from data using the framework of this paper. The results for the lowest level should be ignored as the electron gas is thought to depopulate nonuniformly21 as the gate bias is decreased in this region, causing a breakdown of our analysis. Comparison of results at different temperatures provides a consistency test of the method. The results shown in Fig. 3 axe from the DOS (On/Op) thermally broadened at 2.1 K. We have also measured the DOS at higher temperatures (for instance at 7 K). This spectrum may be simulated by 'artificially" thermally broadening the low temperature results through convolution with the derivative of the Fermi distribution function (for T = 7 K). The results axe nearly identical to the DOS obtained at 7 K at the same magnetic field value, ls'18 The solid curve shown in Fig. 3 is a fit to Lorentzian lineshapes. The expression used is
~.B~I_ g(S) =
h
r,
~=1 n (E - (i + 1/2)hw~) 2 +
r~
(3)
There are five free parameters in the least squares fits, the widths of the first four Landau levels and the zero of energy (which was fit by eye). The widths of higher index Landau levels are forced to be equal to that of the fourth level. The fits obviously work very well, much better than do Gaussian fits even in the case where a background, constant in energy, is included. The results of Figs. 2 and 3 are from sample B (see table I). In samples A and C Lorentzians also fit the data quite well up to fields (around 4 T) at which the exchange enhanced spin splitting becomes an appreciable fraction of the linewidth. At a field of 4 T in sample A, we find that Gaussian fits require a background of half of the zero field DOS to account for the interlevel DOS, while Lorentzians fit the data even better with no background DOS. Lineshapes at higher magnetic field values are treated elsewhere, ls,l* As might be expected, sample B, containing no doping in the blocking barrier, has the naxrowest lines, whereas sample C, with the most doping, contains the widest. Although the Landau level DOS peak widths observed in our samples are the same or narrower than those of past experiments, the magnetic field dependence of our the linewidths is qtfite different. 8,8 In order to explore the field and concentration dependence of the Landau level width, we have plotted in Fig. 4 the width versus electron concentration, for the different values of the magnetic field at which the Landau level maxima occur, for data from sample A. First, it is evident from this figure that there is no significant variation in width with field. Second, there is a significant decrease in width with increasing electron concentration. This is qualitatively consistent with work showing an increase
Vol. 81, No. 10
of mobility with increasing concentration, described by a power law with exponent between 1.1 and 1.7. 22 The fit shown in Fig. 4 is a power law with exponent -0.28. The magnitude of this number is clearly smaller than that expected from a strict interpretation of the linewidth as proportional to the inverse of the mobility. However, a strict correspondence is not expected 2s because of the different relative contributions of small and large angle scattering to sample mobility and to the inverse scattering time which is important in determining the level width. Similar independence of the linewidths on field strength occurs in sample B as well; we have data only at 4 T in sample C. In summary, we have developed a new technique which determines the Landau level density of states as a function of Fermi energy in a 2d electron gas. There are two essential results from our study of three different samples. First, Landau levels are fit well with Lorentzian lineshapes independent of the doping configuration in our samples. Second, the linewidths of Landau level DOS peaks axe independent of magnetic field. Acknowledgment- We thank S.L. Wright and M. Heiblum, both of I.B.M. Corporation, and L.Pfeiffer of AT&T for providing the wafers used in this work. We are grateful to J. Lebens for work in the initial design of samples. The use of the National Nanofabrication Facility at Cornell was essential to the fabrication of the samples, and the Materials Science Center Computer Facility enabled the analysis of the extensive data set. This work was principally supported by the Semiconductor Research Corporation, Grant No. 90-SC-069.
m. > E~.
' ~'~
'
'
o'2T
2 Ix.
x
e- ,--: 0
~o
~..-: 0 _J
E~ ed L_ 0
)
i
i
L
i
i
1
2
3
4
5
6
Landau level Peek Density (1011 c m -2)
Figure 4. Widths of Landau levels plotted as a function of the density at which the Landau level peak occurs for magnetic fields of 2, 3 and 4 T. The solid curve is a power law fit described in the text. The figure suggests that the Landau level widths are independent of magnetic field strength.
Vol. 81, No. 10
LANDAU LEVEL DENSITY OF STATES
825
REFERENCES
1. A.B. Fowler, F.F. Fang, W.E. Howard, and P.J. Stiles, Phys. Rev. Lett. 16, 901 (1966). 2. I.V. Kukushkin , S.V. Meshkov, and V.B. Timofeev, Sov. Phys. Usp. 31,511, (1988). 3. B.I. Halperin, Phys. Rev. B 25, 2185 (1982). 4. E. Gornik, R. Lassnig, G. Strasser, H.L. StSrmer, A.C. Gossard, and W. Wiegmann, Phys. Rev. Lett. 54, 1820 (1985). 5. J.K. Wang, J.H. Campbell, D.C. Tsui, and A.Y. Cho, Phys. Rev. B 38, 6174 (1988). 6. J.P. Eisenstein, H.L. StSrmer, V. Narayanmurti, A.Y. Cho, A.C. Gossard, and C.W. Tu, Phys. Rev. Lett. 55,875 (1985). 7. T.P. Smith, B.B. Goldberg, P.J. Stiles, and M. Heiblum, Phys. Rev. B 39., 2696(1985). 8. V. Mosser, D. Weiss, K. vonKlitzing, K. Ploog, and G, Weimann, Solid State Comm. 58, 5 (1986). 9. S.V. Kravchenko, D.A. Rinberg, A.G. Semenchinsky, and V.M. Pudalov, Phys. Rev. B42, 3741
(1990). 10. T. Ando and Y. Uemura, J. Phys. Soc. Jpn. 36,
959 (1974). 11. F. Wegner, Z. Phys. B 51, 279 (1985). 12. S. Das Sarma and X.C. Xie, Phys. Rev. Lett. 61, 738 (1988). 14. R.C. Ashoori, J.A. Lebens, N.P. Bigelow, and R.H. Silsbee, Phys. Rev. Lett. 64, 681 (1990). 13. T.P. Smith III, W.I. Wang, and P.J. Stiles, Phys. Rev. B 34, 2995 (1986). 15. R.C. Ashoori, Ph.D. thesis, Cornel] University (1991). 16. R.C. Ashoori et al. (to be published). 17. A.L. Efros, Solid State Commun. 65, 1281 (1988). 18. F. Stern and S.E. Lanx, private communication. 19. J.A. Lebens, Ph.D. thesis, Cornell University (1988). 20. G.A. Baraff and Joel A. Appelbaum, Phys. Rev. B 5,475 (1972). 21. John A. Nixon and John H. Davies, Phys. Rev. B 41, 7929 (1990). 22. Kazuhiko Hirakawa and Hiroyuki Sakaki, Phys. Rev. B 33, 8291 (1986). 23. S. Das Sarma and Frank Stern, Phys. Rev. B 32, 8442 (1985).
Solid State Communications, Vol. 81, No. 10, pp. 821-825, 1992. Printed in Great Britain.
0038-1098/9255.00+.00 Pergamon Press plc
THE LANDAU LEVEL DENSITY OF STATES AS A FUNCTION OF FERMI ENERGY IN THE TWO DIMENSIONAL ELECTRON GAS R.C. Ashoori* and R.H. Silsbee
Laboratory of Atomic and
Solid State Physics, Cornell University, Ithaca, New York 14853, USA
(Received 11 December 1991 by A. Pinczuk) We have made quantitative determinations of the density of states (DOS) of Landau levels in a 2d system whose density can be varied using a gate bias. A novel technique employs two normalization conditions to extract the DOS as a function of Fermi energy in the 2d electron gas from capacitance data. No knowledge of sample parameters is required except for the sample area. In contrast with previous results, we find that Lorentzian lineshapes give excellent fits to the Landau level structure without need for added background and with llnewidths that are independent of magnetic field strength.
Since the early magnetoconductance work of Fowler et. al.1 in silicon MOSFETs, and especially after the discovery of the quantum Hall effect, there has been a great effort to understand the Landau level density of states (DOS) in two dimensional (2d) systems. 2 A clear picture of the Landau level DOS should be an aid for understanding features in the quantum Hall effect. The existence of an interlevel density of localized states is thought to be of critical importance for the occurrence of Hall plateaus, s There have been several measurements of the Landau level DOS with experiments using specific heat, 4,5 magnetization,B and capacitance. 7-9 Although a clear picture of the Landau level DOS has not emerged, several of these experiments suggest that the Landau level DOS can be described by a Gaussian shape (sometimes with an added ad hoc constant background) whose width depends on magnetic field. Previous experimental results 6 have indicated a B 1/2 dependence of the width of lines in agreement with the short range scattering model of Ando and Uemura. ~° Theoretical arguments have predicted elliptical lineshapes, 1° Gaussian lineshapes, 13 or lines whose shapes depend in a complicated way on the structure of the sample in which the 2d electron gas exists. 12 This Solid State Communication reports a series of capacitance measurements which give significantly different results. Specifically, Lorentzian lineshapes give excellent fits to the Landau level DOS in three samples with different doping profiles, and the widths of these lines are independent of magnetic field. We have developed a new technique for determination of the Landau level DOS. The method allows transformation of capacitance data directly into a quantitative measurement of the density of states without need
for any sample parameters (or even the dielectric constant of the medium) except for the lateral area of the sample which is well known. Even the effects of band bending in the top and bottom gates of the sample are automatically included in the analysis. Previous capacitance determinations of the Landau level DOS have been susceptible to large errors because of uncertainties in the values of sample parameters, as noted by Smith et. al. 13 Further, our experiments are done in samples in which the 2d electronic density can be varied by means of a gate bias. Our analysis explicitly yields the Landau level DOS as a f u n c t i o n of Fermi energy in the 2d electron gas. This allows immediate comparison of the DOS results discussed here with models that describe the DOS as a function of energy in the 2d gas, in contrast with other experiments which typically measure the DOS at the Fermi energy for a constant 2d electron density as the magnetic field strength is swept. Fig. 1 shows a conduction band edge diagram for one of the three samples used in the present study and is typical of all the samples. The samples are grown on n + GaAs conducting substrates using molecular beam epitaxy. They are of the same type that we have used in tunneling experiments. 14 Grown on top of the substrate are a GaAs undoped spacer layer, an A1GaAs tunnel barrier, a 150,~ wide GaAs quantum well, a thick AIGaAs "blocking barrier", and finally a heavily n doped GaAs top contact layer (gate). In some samples the blocking barrier, which allows no electrical conduction, contains a meg'ion of Si doping. Ohmic contact is made to the gate and the substrate. Table I lists various growth parameters for the samples used. The capacitance of these devices is a function of frequency. In the simplest picture, only valid for infinite DOS in the well, charge transfer between the quantum well and the substrate can be described in terms of a parallel R C circuit, with R the tunneling resistance and
* Present address: AT&T Bell Laboratories, Murray Hill, N.I 07974. 821
822
LANDAU LEVEL DENSITY OF STATES
Vol. 81, No. 10
Table 1. Growth parameters for the saxnples.
A
30
160
1550
Si dopants (from well edge) (/~) 100-200
B
30
133
800
no dopants
no dopants
C
150
150
800
150-500
6x1017
Sample
GaAs Spacer layer (~)
Tunnel Blockin~ Barrier (/~) Barrier (A)
C the capacitance between the 2d electron gas and the substrate. At frequencies small compared to 1/27rRC there is sufficient time for equilibrating charge to traverse the tunnel barrier during one half cycle of the measuring voltage, while at high frequencies little charge is transferred. We refer to the low and high frequency limiting values of the capacitance as CIo~ and Chigh respectively. In our experiments, the device capacitance is measured as a function of frequency over the range from 15 Hz to 30 kHz. This is done for various values of the gate bias, magnetic field applied perpendicular to the plane of electrons in the well, and temperature. Fits, based on a model described elsewhere] sAs are made to the capacitance to extract Crow and Chig~. The range of frequencies over which we measure the capacitance is sufficiently wide that at the lowest and highest frequencies the capacitance has essentially obtained its limiting values. Cto~ depends on the DOS in the quantum well, but Chigh does not. If the DOS in the well were infinite, G'tow would simply be determined by the sample area, the dielectric constants, and the spacing between the 2d plane and the gate. In contrast, if the DOS were equal to zero, there could be no charge transfer between the substrate and the well, and Crow would be equivalent
to Chigh. As the density of states in the well is varied between zero and infinity, Ctoto varies monotonically and predictably from a lower limit at zero DOS equal to Chigh to an upper limit for infinite DOS. Fig. 2 shows Clow and Chiah for a sample with no doping in the blocking barrier as a function of gate bias on the device. The results axe for 2 T applied magnetic field and a temperature of 2.l K. In the region below -80 mV in the figure, the quantum well is depleted of electrons and the DOS in the well is zero. In this case C~o,,, and Chigh have the same value. As the gate bias is increased, electrons begin to enter the well (the bound state energy in the well drops below the Fermi energy in the substrate), the DOS increases from zero, and Cto,o increases sharply. As the gate bias is increased further, Cto~, begins to oscillate as a function of gate bias. These oscillations arise due to the Landau level DOS in the well, with peaks at the Landau level centers and minima between levels. Ctow oscillates about its zero field value. Notice that Chigh contains no oscillations, as expected since Chigh is insensitive to the DOS in the well. Two equations provide interpretation of these results. T M One gives the "lever-arm" which relates the applied gate voltage V~t~ to the bound (ground) state energy in the well Ubou,,d measured relative to the fixed
"l Blocking Barrier
Gale
Well
Tunnel
Dopant Concentration (cm-3) 5x1017
'
'
'1
'
Subslrate
Barrier
1 I 1' 1
x x ×
C" v~
x x
0 E 0
x x x
X"
I
x x
"/".................. L ~
x × x × ×
. . .. .. .. .. .. ..
,~t'#
-100
0
100
i
i
200
300
t
400
500
600
Gate Voltoge (rnV)
Figure 1. The essential structure of the samples. Electron transfer through the tunnel barrier brings the electron gases in the substrate and the quantum well into equilibrium. The density of electrons in the quantum well may be varied through the application of a gate bias (Vg~t, in the figure).
Figure 2. Capacitances Crow and Chigh for sample B at 2.0 T and 2.1 K. The two dashed vertical lines are limits of integration for the normalization condition which :letermines Cg. . . . We integrate over two Landau levels :ather than one for higher precision.
Vol. 81, No. 10
LANDAU LEVEL DENSITY OF STATES
Fermi energy in the substrata (which, because of equilibration through the tunnel barrier, is the same as the Fermi energy in the quantum well). The lever-arm is given by
dUbo~,,~d d(eVga,e)
Crow -
C,eo,.,
C,o~ -1)(1 (C--~igh
rlCgeom). (1) Ae 2
Here e is the magnitude of the electron charge. The other equation is for the density of states g which is obtained through • Clow Ae=g = Cgeom( ~
_ 1) d(eVg~,,,)
dUbo~,~d"
(2)
A is the area of the sample which is known to within 1%. These equations involve two model parameters. Cg~om is the capacitance of the quantum well to the substrata in a model in which the DOS is infinite and the well charge is taken to lie at the center of charge of the real charge distribution in the well. 7/is a parameter describing the variation in bound state energy of the well associated with the details of the distribution of the electron density in the well as it is filled. 0 requires a brief explanation. Our analysis begins with a simplified model which considers electrons in the quantum well to lie in a sheet located at the mean position of charges in the well. The energy of the well bound state is initially taken to be a fixed energy E0 greater than conduction band edge energy U~, calculated in the sheet charge model, at this position, r] is the coefficient of a term qaw (where aw is the sheet electron number density in the well) which is subtracted from Uw + Eo to yield Ubo~,d, a more accurate result for the bound state energy. The r/a~0 term corrects the electrostatics in our analysis for the actual configuration of distributed charge in the well and also includes adjustments to the bound state energy for its quantum mechanical variation arising from changes in curvature of the band bottom of the well as the well is filled with electrons. We show elsewhere 15,16 that the rigidity of the shape of the electronic wavefunction in the well as the well is filled and the adequacy of first order perturbation theory in calculating changes to the bound state energy justify the inclusion of only this linear order correction. The exchange-correlation interaction9,17 also decreases E0 as the well is filled, and 7/absorbs the linear order variation in E0 as a function of density due to this effect. If reliable values of 77and Cg~o,, can be deduced from sample characterization and theoretical modeling then Eqs. 1 & 2 give the desired quantities. Unfortunately, modeling of the sample typically will no~ give sufficient accuracy in determination of these critical parameters to allow for confidence in the DOS results. Our method requires no such sample modeling. Two normalization conditions are available to determine C0.... and 7/ through knowledge of the experimentally measured quantities, Clow and Chig h. The degeneracy of a Landau level, when the sample is placed in magnetic
823
field perpendicular to the quantum well, is the number of flux quanta threading the sample per unit area times the spin degeneracy of two, or 2Be/h. Assuming a fixed value of Cg~om as the density in the quantum well changes, Eq. 2 may be integrated between adjacent minima of the measured quantity CIow/Chig h (corresponding to minima in the DOS) to determine Cg. . . . It is also known that the lever-arm, when integrated in Vg~,t~over a Landau level, must give tuzc/e, where wc is the cyclotron frequency. 7/is then determined by integrating Eq. 1 over gate voltage between adjacent Landau level peaks in the DOS. With Cg~om and r/ determined by these conditions, Eq. 2 gives the DOS directly from the data and knowledge of the sample area. With the inclusion of a small variation of Cg. . . . linear in gate bias and determined from examination of data taken in zero magnetic field, to account for the small changes with gate voltage of the values of 0 and Cocom,15,x6 this procedure provides a direct translation of the data in Fig. 2 to the Landau level DOS as a function of energy. The magnitude of this adjustment is in rough accord with computer simulations determining the variation in the mean position of electrons in the well, is calculations19 of the shift of the mean position of the substrata charge using the Thomas-Fermi approach of Baraff and Appelbaum, 2° and the expected variation in r/. We point out that the value of Cg~o,, extracted is in good agreement with expectations for the distance separation between the quantum well and substrata charge densities. In fields up to 4 T, the method
i
E~ 0 3 c~
o
-hOJc--
-9
-6
-3
-hwc-
0
-hwc-
3
6
Fermi Energy in Quantum Well (meV)
Figure 3. The circles are the DOS (On/O#) vs. Fermi energy in sample B at 2.0 T and 2.1 K, extracted using the analysis procedure described in the paper, from the data presented the data of Fig. 2. The vertical lines drawn are hcoc apart in energy. Note, one of the two unknown sample parameters in our model was adjusted so as to keep two of the peaks h~zc apart. The fact that other peaks also fall hwc apart affirms the consistency of our model. The solid curve is a fit to Lorentzian lineshapes. The fitting parameters are: Fj=1.47 rneV, F2=0.81 meV, F.~=0.74 meV, and P4=0.68 mcV.
824
LANDAU LEVEL DENSITY OF STATES
yields the same values of Cg~o,n independent of field strength. Using this procedure we plot in Fig. 3 the DOS based on the capacitance data of Fig. 2. We stress that the results of Fig. 3 have been obtained entirely from data using the framework of this paper. The results for the lowest level should be ignored as the electron gas is thought to depopulate nonuniformly21 as the gate bias is decreased in this region, causing a breakdown of our analysis. Comparison of results at different temperatures provides a consistency test of the method. The results shown in Fig. 3 axe from the DOS (On/Op) thermally broadened at 2.1 K. We have also measured the DOS at higher temperatures (for instance at 7 K). This spectrum may be simulated by 'artificially" thermally broadening the low temperature results through convolution with the derivative of the Fermi distribution function (for T = 7 K). The results axe nearly identical to the DOS obtained at 7 K at the same magnetic field value, ls'18 The solid curve shown in Fig. 3 is a fit to Lorentzian lineshapes. The expression used is
~.B~I_ g(S) =
h
r,
~=1 n (E - (i + 1/2)hw~) 2 +
r~
(3)
There are five free parameters in the least squares fits, the widths of the first four Landau levels and the zero of energy (which was fit by eye). The widths of higher index Landau levels are forced to be equal to that of the fourth level. The fits obviously work very well, much better than do Gaussian fits even in the case where a background, constant in energy, is included. The results of Figs. 2 and 3 are from sample B (see table I). In samples A and C Lorentzians also fit the data quite well up to fields (around 4 T) at which the exchange enhanced spin splitting becomes an appreciable fraction of the linewidth. At a field of 4 T in sample A, we find that Gaussian fits require a background of half of the zero field DOS to account for the interlevel DOS, while Lorentzians fit the data even better with no background DOS. Lineshapes at higher magnetic field values are treated elsewhere, ls,l* As might be expected, sample B, containing no doping in the blocking barrier, has the naxrowest lines, whereas sample C, with the most doping, contains the widest. Although the Landau level DOS peak widths observed in our samples are the same or narrower than those of past experiments, the magnetic field dependence of our the linewidths is qtfite different. 8,8 In order to explore the field and concentration dependence of the Landau level width, we have plotted in Fig. 4 the width versus electron concentration, for the different values of the magnetic field at which the Landau level maxima occur, for data from sample A. First, it is evident from this figure that there is no significant variation in width with field. Second, there is a significant decrease in width with increasing electron concentration. This is qualitatively consistent with work showing an increase
Vol. 81, No. 10
of mobility with increasing concentration, described by a power law with exponent between 1.1 and 1.7. 22 The fit shown in Fig. 4 is a power law with exponent -0.28. The magnitude of this number is clearly smaller than that expected from a strict interpretation of the linewidth as proportional to the inverse of the mobility. However, a strict correspondence is not expected 2s because of the different relative contributions of small and large angle scattering to sample mobility and to the inverse scattering time which is important in determining the level width. Similar independence of the linewidths on field strength occurs in sample B as well; we have data only at 4 T in sample C. In summary, we have developed a new technique which determines the Landau level density of states as a function of Fermi energy in a 2d electron gas. There are two essential results from our study of three different samples. First, Landau levels are fit well with Lorentzian lineshapes independent of the doping configuration in our samples. Second, the linewidths of Landau level DOS peaks axe independent of magnetic field. Acknowledgment- We thank S.L. Wright and M. Heiblum, both of I.B.M. Corporation, and L.Pfeiffer of AT&T for providing the wafers used in this work. We are grateful to J. Lebens for work in the initial design of samples. The use of the National Nanofabrication Facility at Cornell was essential to the fabrication of the samples, and the Materials Science Center Computer Facility enabled the analysis of the extensive data set. This work was principally supported by the Semiconductor Research Corporation, Grant No. 90-SC-069.
m. > E~.
' ~'~
'
'
o'2T
2 Ix.
x
e- ,--: 0
~o
~..-: 0 _J
E~ ed L_ 0
)
i
i
L
i
i
1
2
3
4
5
6
Landau level Peek Density (1011 c m -2)
Figure 4. Widths of Landau levels plotted as a function of the density at which the Landau level peak occurs for magnetic fields of 2, 3 and 4 T. The solid curve is a power law fit described in the text. The figure suggests that the Landau level widths are independent of magnetic field strength.
Vol. 81, No. 10
LANDAU LEVEL DENSITY OF STATES
825
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