Several years ago, Kravchenko *et al.* [ @1] observed that the resistivity ($\rho$) of the high mobility two-dimensional (2D) electron gas in their Si metal-oxide-semiconductor field-effect transistor (MOSFET) samples, decreased by almost an order of magnitude when they lowered their sample temperature ($T$) below about 2K. Their observation of such a metallic behavior contradicts the scaling theory of localization which [@2] predicts that, in the absence of electron-electron interaction, all states in 2D are localized in the $T\rightarrow$0 limit and that only an insulating phase characterized by an increasing $\rho$ with decreasing $T$ is possible at low $T$. They studied the $T$ dependence of $\rho$ as a function of the 2D carrier density ($p$) and demonstrated from the temperature coefficient (d$\rho$/d$T$) of $\rho$ a clear transition from the metallic to an insulating behavior at a “critical” density, $p_c$. This apparent metal-insulator transition (MIT) has since been reported in other low disorder 2D systems [@3], and it appears to be a general phenomenon in low disorder dilute 2D systems where the Fermi energy is small and $r_s$ (the ratio of Coulomb interaction energy to Fermi energy) is large ($\agt$10).  To date, despite the large number of experimental [@3; @4; @5; @6; @7; @8; @9; @10] and theoretical [@11; @12] papers on this zero field MIT in the literature, there is still no consensus on the physics and the mechanisms behind this metallic behavior. 

Two factors that strongly influence this MIT have become apparent from the more recent experiments.  First, the application of a magnetic field parallel to the 2D system ($B_{\parallel}$) induces a drastic response [@8; @9; @10].  A giant positive magetoresistance is observed in both the metallic and the insulating phases, varying continuously across the transition.  In the case of Si MOSFET’s, Simonian *et al.* [ @8] have made a detailed study of the temperature and electric field dependences on the magnetoresistance and concluded that “in the $T\rightarrow$0 limit the metallic behavior is suppressed by an arbitrarily weak magnetic field”.  Since $B_{\parallel}$ couples only to the carrier’s spin and does not affect its orbital motion, the spin degree of freedom must play a crucial role in the electronic processes that give rise to transport in both phases. 

The second factor that has become increasingly clear is the importance of the role played by disorder.  A close examination of all 2D systems that show the metallic behavior and thus the MIT as the carrier density is reduced reveals that $p_c$ is lower in systems with a higher mobility (i.e., a lower carrier scattering rate) [@5].  Typically, the 2D electron system (2DES) in a high quality Si MOSFET has a peak mobility of $1\times10^{4}-5\times10^{4} cm^{2}/Vs$ and $p_c\approx1\times10^{11} cm^{-2}$. On the other hand, the 2D hole system (2DHS) in GaAs/AlGaAs heterostructures, which has a comparable effective mass ($m^{*}$) at low densities, usually has a peak mobility of about 10 times higher and $p_c\approx1\times10^{10} cm^{-2}$. It is clear that $p_c$ decreases monotonically with decreasing disorder in the 2D system.  In the ideal clean limit, it is well known that the Wigner crystal is the ground state in the low $p$ limit.  The estimated critical density for Wigner crystallization is approximately $2\times10^9 cm^{-2}$ for the 2DHS in GaAs, using $m^{*}=0.18m_{e}$ ($m_e$ being the free electron mass) and a value of 37 for $r_s$ at the crystallization [@13; @14]. 

We have recently investigated the transport properties of a 2DHS in the GaAs/AlGaAs heterostructure, which has an unprecedentedly high peak mobility of $7\times10^{5} cm^{2}/Vs$, and observed a zero field MIT at $p_c=7.7\times10^{9} cm^{-2}$ [@5].  The mobility of this 2DHS is over 25 times that of the 2DES in the Si MOSFET whose parallel magnetic field response has been most extensively studied in references 8 and 9.  In view of the fact that it is not yet clear what specific role the spins play in the two transport regimes and how the small amount of random disorder in high quality 2D systems influences the MIT, we have systematically studied the effect of a $B_{\parallel}$ on the transport in this high quality 2DHS.  We find that for $p>p_{c}$, the metallic behavior persists to our lowest $T$ of 50mK until $B_{\parallel}$ reaches a well defined “critical” value $B_{\parallel}^{c}$, beyond which the 2DHS shows an insulating behavior.  At $B_{\parallel}^{c}$, $\rho$ is independent of $T$. The nonlinear I-V characteristics across this $B_{\parallel}$ induced transition is found to be the same as those across the zero field MIT.  Below, we describe in more detail the changes in the transport properties of the 2DHS under the influence of a $B_{\parallel}$ and report our observation of this $B_{\parallel}$ induced MIT. 

We used the 2DHS created in a Si modulation doped GaAs/Al$_{x}$Ga$_{1-x}$As heterostructure grown on the (311)$A$ surface of an undoped GaAs substrate by molecular beam epitaxy.  The samples were Hall bars along the \[$\overline {2}$33\] crystalographic direction, and the measurements were made using a dilution refrigerator in the $T$ range from 50mK to 1.1K and under $B_{\parallel}$ up to 14T. The hole density was tuned by a back gate in the range $5.7\times10^{9}<p<4.1\times10^{10} cm^{-2}$.

In Fig.  1(a), the $T$ dependence of $\rho$ in the zero field metallic phase is shown on a semilog plot for a hole density $p=3.7\times10^{10} cm^{-2}$ at several different $B_{\parallel}$’s.  The bottom trace taken at $B_{\parallel}=0$ clearly shows a positive $d\rho/dT$, which is characteristic of metallic-like transport.  This metallic behavior is found to persist to our lowest $T$ of 50mK in a magnetic field of up to about 4T. As $B_{\parallel}$ increases from zero, the strength of the metallic behavior measured by the total change in $\rho$ from about 1K to 50mK weakens progressively, and for $B_{\parallel}\geq4.5$T $d\rho/dT$ becomes negative.  We take this negative $d\rho/dT$ as an indication that the 2DHS is insulating, and phenomenologically identify the two distinct transport regimes as a “metallic phase” and an “insulating phase”.  It is clear from the figure that there exists a “critical” field $B_{\parallel}^{c}$ near 4T where $\rho$ becomes $T$ independent, separating the metallic and insulating phases.  Another way of demonstrating the existence of a well defined $B_{\parallel}^{c}$ is to plot $\rho$ against $B_{\parallel}$ at several different $T$’s.  Such a plot is shown in Fig.  1(b) for $p=1.5\times10^{10} cm^{-2}$. In this plot, the crossing point marked by the arrow defines $B_{\parallel}^{c}$. This is the direct consequence of the fact that $\rho$ decreases with decreasing $T$ in the metallic phase ($B_{\parallel}<B_{\parallel}^{c}$), increases in the insulating phase ($B_{\parallel}>B_{\parallel}^{c}$), and is independent of $T$ at $B_{\parallel}^{c}$.

Across the zero field MIT, the differential resistivity ($dV/dI$) is known to show an increase with increasing voltage ($V$) in the metallic phase and a decrease in the insulating phase [@4; @5].  In Fig.  1(c), the $dV/dI$ measured at 50mK across the $B_{\parallel}$ induced MIT for $p=3.7\times10^{10} cm^{-2}$ is shown at similar $B_{\parallel}$’s as in Fig.  1(a).  It is clear that in the metallic phase ($B_{\parallel}<4.2$ T) $dV/dI$ increases as $\vline$$V$$\vline$ increases, and in the insulating phase ($B_{\parallel}>4.2$T) it decreases.  At $B_{\parallel}=4.2$T which is the $B_{\parallel}^{c}$, $dV/dI$ is constant, implying a linear $I-V$. This result also shows that there is a well defined critical field $B_{\parallel}^{c}$ separating the metallic and insulating phases in the presence of a parallel magnetic field. 

We have measured $B_{\parallel}^{c}$ and the “critical” resistivity ($\rho_{c}$) as a function of $p$ in the $p>p_{c}$ regime for two samples cut from the same wafer, and the results are shown in Fig.  2(a)-(c).  Figures 2(a) and 2(b) show that $B_{\parallel}^{c}$ decreases with decreasing $p$ and approaches zero as $p$ is reduced towards $p_{c}$ where the zero field MIT is observed.  When $B_{\parallel}^{c}$ is plotted against $p-p_{c}$ on a log-log scale (Fig.  2(a)), all data from the two samples form two parallel lines, showing that $B_{\parallel}^{c}\propto(p-p_{c})^{\alpha}$ with $\alpha\approx0.7$ for both samples.  It is interesting to note that Hanein *et al.* [ @7] extracted from the $T$ dependence of $\rho$ in the zero field metallic phase in a previous experiment, an energy scale, $T_{0}$, in the form of an activation energy.  Their $T_{0}$ depends linearly on $p$ in the range $p>2\times10^{10} cm^{-2}$ and extrapolates to zero at $p=0$. We postulate that the magnetic energy at $B_{\parallel}^{c}, 
g\mu_{B}B_{\parallel}^{c}$ ($g$ being the $g$-factor of holes in GaAs and $\mu_{B}$ the Bohr magneton), is equivalent to their $k_{B}T_{0}$ ($k_{B}$ is the Boltzmann constant), and compare the $p$ dependences of $B_{\parallel}^{c}$ and $T_{0}$ by replotting $B_{\parallel}^{c}$ against $p$ on a linear scale in Fig.  2(b).  We find that our data from both samples in the range $p>2\times10^{10} cm^{-2}$ fall on straight lines that extrapolate to zero at $p=0$, and therefore the dependence of $B_{\parallel}^{c}$ on $p$ is similar to that of $T_{0}$ on $p$ in the same $p>2\times10^{10} cm^{-2}$ range.  If we equate our $g\mu_{B}B_{\parallel}^{c}$ with their $k_{B}T_{0}$ at $p>2\times10^{10} cm^{-2}$, we obtain a $g$-factor of 0.1, which is of the same order as the hole $g$-factor of a $100\AA$ wide GaAs/AlGaAs quantum well [@15].  However, we are not able to distinguish whether the $p$ dependence of $B_{\parallel}^{c}$ for $p>2\times10^{10} cm^{-2}$ is indeed linear or of the power law form because the density range covered in our measurements is small. 

The “critical” 2D resistivity $\rho _{c}$ at the transition depends strongly on $B_{\parallel}^{c}$ and therefore on $p$. At $p=p_{c}$, where $B_{\parallel}^{c}=0$, $\rho _{c}$ is of the order of one resistance quantum, $\rho_{Q}=h/e^{2}$ (where $h$ is Plank’s constant and $e$ the electron charge).  For $p>p_{c}$, $\rho_{c}$ decreases steeply as $B_{\parallel}^{c}$ increases and drops to $\sim0.03\rho_{Q}$ at $B_{\parallel}^{c}\approx7$T, as shown by the solid circles in Fig.  3.  Fig.  2(c) shows the $\rho_{c}$ data from both samples as a function of $p-p_{c}$, and it is clear that for $p-p_{c}>2\times10^{9} cm^{-2}$ $\rho_{c}$ decreases exponentially with increasing $p$. This strikingly strong dependence of $\rho_{c}$ on $p$ is not anticipated within the MIT framework.  It suggests that the observed insulating behavior for $B_{\parallel}>B_{\parallel}^{c}$ cannot be the result of thermally activated processes in a simple Anderson type of insulator.  However, it is reminiscent of the magnetic field driven superconductor-insulator transition reported by Yazdani and Kapitulnik [@16] in their experiments on thin films of amorphous MoGe, where similar decrease in critical resistivity with increasing critical $B$ field is observed.  They have attributed this lack of universality in their critical 2D resistivity to the presence of conduction by unpaired electrons.  In this context, we should also note that Phillips *et al.* [ @11] have proposed that the metallic behavior observed in high mobility 2D systems is that of a superconductor, and the existence of a critical $B$ field is to be expected.  However, $\rho$ in the metallic regime is known to saturate to a finite value instead of vanishing as $T\rightarrow0$, and the relation of the metallic behavior to superconductivity is not known at present. 

Now, we turn to the discussion on the overall in-plane magnetoresistance.  In Fig.  3, the in-plane magnetoresistance in the zero $B$ field insulating phase ($p<p_{c}$) is shown as open circles and in the metallic phase ($p>p_{c}$) as solid lines.  Regardless of whether the zero field transport is metallic or insulating, we observe a strong positive magnetoresistance.  According to the $B_{\parallel}$ dependence of $\rho$, we can divide the entire $\rho-B_{\parallel}$ plane into two regimes: a low field regime and a high field regime.  In the low field regime, we find that the magnetoresistance is well described by $\rho=\rho_{0}exp(B_{\parallel}^{2}/B_{0}^{2})$, where $\rho_{0}$ and $B_{0}$ are the fitting parameters.  The value of $B_{0}$ is shown as the solid circles in Fig.  4 as a function of $p$. $B_{0}$ decreases as $p$ is reduced towards $p_{c}$ (marked by the arrow in Fig.  4) reflecting that the $B_{\parallel}$ dependence of $\rho$ becomes stronger.  However, it is clearly visible in Fig.  4 that $B_{0}$ saturates to a constant value of $\sim3$T as the 2DHS is brought into the zero field insulating phase.  The $B_{\parallel}$ induced transition occurs in this low field regime, and the measured $B_{\parallel}^{c}$’s are marked by the solid circles in Fig.  3 (the dashed line is a guide to the eye).  As the magnetic field is increased beyond $B_{\parallel}^{*}$ (which is indicated by the dotted line in Fig.  3) into he high field regime, the dependence of $\rho$ on $B_{\parallel}$ changes.  In this regime, the magnetoresistance is of the form $\rho =\rho _{1}exp(B_{\parallel}/B_{1})$, where $\rho _{1}$ and $B_{1}$ are the fitting parameters.  As shown by the open circles in Fig.  4, the $p$ dependence of $B_{1}$ is similar to that of $B_{0}$; $B_{1}$ also decreases as $p$ is reduced towards $p_{c}$ and saturates to $\sim 1.5$T at $p<p_{c}$. While the overall magnetoresistance evolves smoothly as $p$ is changed across $p_{c}$, it is obvious from Fig.  3 that $B_{\parallel}^{*}$ (the boundary separating the low field and high field regimes) decreases with decreasing $p$ in the zero field metallic phase, but is independent of $p$ in the insulating phase.  It is interesting to note that all three characteristic fields for the in-plane magnetoresistance, $B_{0}$, $B_{1}$, and $B_{\parallel}^{*}$, become independent of $p$ when $p$ is reduced below $p_{c}$.

The low field magnetoresistance at $B_{\parallel}<B_{\parallel}^{*}$ is very similar to that observed in the Si MOSFET’s [@8; @9; @10], where a positive magnetoresistance at low fields is followed by a saturation at high fields.  Mertes *et al.* [ @9] have interpreted the low field positive magnetoresistance in Si MOSFET’s by the hopping model of Kurobe and Kamimura [@17], in which hopping becomes more difficult as more spins get aligned with $B_{\parallel}$. This interpretation is supported by the observation of Okamoto *et al.* [ @10] that the field where magnetoresistance starts to saturate coincides with the field expected for complete spin alignment.  Such a hopping model can also be applied to explain our data in the $p<p_{c}$ insulating regime.  However, for $p>p_{c}$, transport is metallic and hopping is not relevant; some other mechanisms involving spins must be operative. 

The exponential divergence of the magnetoresistance observed at high fields ($B_{\parallel}>B_{\parallel}^{*}$), on the other hand, has not been observed in other 2D systems before, and cannot be explained by existing theoretical models.  The model by Lee and Ramakrishnan [@18] for a weakly disordered system, of which the in-plane magnetoresistance arises from spin splitting, predicts a logarithmic divergence.  In the hopping model by Kurobe and Kamimura, the magnetoresistance is expected to saturate when all spins are aligned.  In our data, the exponential dependence of $\rho$ on $B_{\parallel}$ is observed up to our highest field of 14T, and $B_{0}$, as seen in Fig.  4, varies continuously with $p$ across $p_{c}$. Also, it appears that the influence of the parallel magnetic field on the energy structure of our 2DHS is not the cause of such strong but simple $B_{\parallel}$ dependence in both transport regimes.  It is possible that this exponential divergence of the magnetoresistance is a phonomenon characteristic of new electronic processes in the 2D system in its clean limit. 

We thank R. Bhatt, P. Phillips, M. Hilke, S. Papadakis, and Y. Hanein for fruitful discussions.  This work is supported by the NSF. 

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Fig.  1. ( a) $T$ dependence of $\rho$ at $p=3.7\times10^{10} cm^{-2}$ and at $B_{\parallel}$=0, 2, 3, 3.5, 4, 4.5, 5, 5.5, 6, and 7T from the bottom. ( b) $\rho$ is plotted as a function of $B_{\parallel}$ at $p=1.5\times10^{10} cm^{-2}$ and at five different temperatures.  The crossing point defines the $B_{\parallel}^{c}$ at 2.1T as marked by the arrow. ( c) Differential resistivity ($dV/dI$) is plotted against $V$ at 50mK and $p=3.7\times10^{10} cm^{-2}$, at magnetic fields, $B_{\parallel}$=0, 2, 3, 3.5, 4.2, 5, 5.5, 6, and 7T from the bottom. 

Fig.  2. ( a) $B_{\parallel}^{c}$ is plotted against $p-p^{c}$. The solid and open circles are for two samples cut from the same wafer. ( b) $B_{\parallel}^{c}$ vs $p$ in a linear scale.  The dotted lines are to indicate that $B_{\parallel}^{c}$ is approximately linear in the range $p>2\times10^{10} cm^{-2}$ extrapolating to zero at $p=0$. (c) $\rho^{c}$ is shown as a function of $p-p^{c}$.

Fig.  3.  The $B_{\parallel}$ dependence of $\rho$ is shown at 50mK at hole densities, from the bottom.  4.11, 3.23, 2.67, 2.12, 1.63, 1.10, 0.98, 0.89, 0.83, 0.79, 0.75, 0.67, and 0.57$\times10^{10} cm^{-2}$. The solid lines are for $p>p^{c}$ and the open circles for $p<p^{c}$. The colid circles denote experimentally determined $B_{\parallel}^{c}$’s, and the dashed line is a guide to the eye.  $B_{\parallel}^{*}$, the boundary separating the low field and the high field regimes, is marked as the dotted line. 

Fig.  4.  $B_{0}$ and $B_{1}$, obtained from fitting the data in Fig.  3 in the form $\rho=\rho_{0}exp(B_{\parallel}^{2}/B_{0}^{2})$ for the low field regime and $\rho=\rho_{1}exp(B_{\parallel}/B_{1})$ for the high field regime, are plotted as a function of $p$. The dashed lines are guides to the eye.  The “critical” density $p^{c}$ is marked by the arrow
